# Saturated fusion systems on $p$-groups of maximal class

Valentina Grazian  Department of Mathematics and Applications
University of Milano – Bicocca
Via Roberto Cozzi 55, 20125 Milano
Italy<EMAIL_ADDRESS>and  Chris Parker  School of Mathematics
University of Birmingham
Edgbaston
Birmingham B15 2TT
United Kingdom<EMAIL_ADDRESS>
###### Abstract.

For a prime number $p$, a finite $p$-group of order $p^{n}$ has maximal class
if it has nilpotency class $n-1$. Here we examine saturated fusion systems on
maximal class $p$-groups and, in particular, we describe all the reduced
saturated fusion systems on these groups.

###### Acknowledgement.

The authors would like to thank the Isaac Newton Institute for Mathematical
Sciences for support and hospitality during the programme _Groups,
representations and applications: new perspectives_ when work on this paper
was undertaken. This work was supported by EPSRC Grant Number EP/R01 146 04/1.

##  1\. Introduction and Main results

Let $p$ be a prime and $S$ be a finite $p$-group. A _fusion system_ on $S$ is
a category which has objects the subgroups of $S$ and morphisms which are
injective group homomorphisms between the subgroups of $S$. A fusion system is
_saturated_ if it satisfies to certain technical axioms which are described in
some detail in Section 5. A saturated fusion system $\mathcal{F}$ is _reduced_
provided $\mathcal{F}=O^{p}(\mathcal{F})=O^{p^{\prime}}(\mathcal{F})$ and
$O_{p}(\mathcal{F})=1$ where these terms have a similar meaning to what they
have in finite group theory. See Section 5 for more details.

All finite groups $G$ determine a saturated fusion system $\mathcal{F}_{S}(G)$
on a fixed Sylow $p$-subgroup $S$ of $G$. In this case the morphisms between
subgroups of $S$ are exactly the restrictions of conjugation maps induced by
elements of $G$. When a saturated fusion system can be constructed from a
group in this way, we say the system is _realizable_. A saturated fusion
system which is not realizable is called _exotic_.

Saturated fusion systems were first introduced under the name _Frobenius
Categories_ by Puig [51], who developed much of their fundamental theory.
Puig’s motivation was to gain a deeper understanding of the local to global
conjectures which are of fundamental importance in modular representation
theory. In particular, saturated fusion systems can be constructed on the
defect group of a $p$-block in much the same way as they are constructed on
Sylow $p$-subgroups. This connection is described by Kessar in [5, Part IV].
Recent developments in this direction include work of Kessar, Linckelmann,
Lynd and Semeraro [36] in which certain numerical conjectures in
representation theory are formulated for fusion systems; in particular, for
exotic fusion systems. This is exciting as, to-date, no saturated fusion
system for an odd prime $p$ has been shown to be exotic without using the
classification of finite simple groups and the research in [36] offers
potential for achieving this without the classification. Finding an
alternative way to show that exotic systems are exotic is listed as [6,
Question 7.7] in the survey by Aschbacher and Oliver. The rediscovery of
Frobenius categories and their christening as saturated fusion systems is
probably traced back to [11] where Broto, Levi and Oliver defined a _centric
linking system_ related to a saturated fusion system. Later Chermak in [15]
demonstrated that each saturated fusion system determines a unique centric
linking system. From the linking system of a saturated fusion system Broto,
Levi and Oliver were able to construct the $p$-completion of its geometric
realization and this is the classifying space associated with the saturated
fusion system. The homotopy properties of these spaces share properties of
$p$-completed classifying spaces of finite groups. Readers are directed to [5,
Part III] for details and motivation for this study. The theory of fusion
systems provides an idealized environment in which to study the $p$-local
structure of a finite group $G$ and, in particular, the structure of
normalizers of non-trivial $p$-subgroups of $G$. This formalism has been
exploited most notably and energetically by Aschbacher in a deep series of
papers aimed at simplifying some parts of the classification of finite simple
groups. This approach is described in the surveys [4, 5, 6]. Other notable
contributions to this goal are [43] in which Oliver determines reduced fusion
systems on $2$-groups in which every subgroup is generated by at most $4$
elements, Andersen, Oliver and Ventura [3] which determines all reduced fusion
systems on $2$-groups of order at most $2^{9}$ and Henke and Lynd [32] where
they consider saturated fusion systems with components related to the Solomon
fusion systems.

The Solomon fusion systems [5, III.6.7] which are supported on a Sylow
$2$-subgroup of $\mathrm{Spin}_{7}(q)$, $q$ odd, are exotic. These are the
only known exotic fusion systems on $2$-groups and are still a fascinating
subject of research [32]. In contrast, there is a large variety of exotic
fusion systems on $p$-groups with $p$ odd and the reason for this is still not
transparent. Ruiz [52] and more recently Oliver and Ruiz [45] have considered
non-abelian simple groups and determined instances where the fusion system
$\mathcal{F}=\mathcal{F}_{S}(G)$ has $O^{p^{\prime}}(\mathcal{F})$ exotic.
These fusion systems, from their very construction are closely related to
finite simple groups. Other exotic systems are constructed as fusion systems
of free amalgamated products, such as those in [10, 16, 50] and these are
often far away from being realized by finite groups in that the $p$-groups are
often not closely related to the Sylow $p$-subgroup of a finite simple group.
Many of these fusion systems are defined on $p$-groups of _maximal class_.
That is, finite $p$-groups of order $p^{n}$ with nilpotency class $n-1$.

The objective of the research presented in this paper is to determine the
saturated fusion systems on $p$-groups of maximal class. In contrast with
other works on fusion systems, we will consider all saturated fusion systems
on the selected class of $p$-groups, not limiting our investigation, for
instance, to reduced saturated fusion systems. The reason for this will become
evident when we discuss the inductive approach to the proof of our theorems.
From now on, assume that $S$ is a $p$-group of order $p^{n}$ and maximal
nilpotency class $n-1$. The smallest interesting possibility is that $n=3$ and
$S$ is an extraspecial $p$-group. This is the case famously handled by Ruiz
and Viruel in the early 2000’s [53] with the, at the time, shocking outcome
that there are three exotic fusion systems on the extraspecial $7$-group of
exponent $7$ and order $7^{3}$. In their work on rank $2$ groups (groups which
have no elementary abelian subgroups of order $p^{3}$ and are not cyclic or
quaternion), Díaz, Ruiz and Viruel [21] examine saturated fusion systems on
the infinite families as classified by Blackburn [8]. The outcome of their
research was the discovery of infinite families of exotic fusion systems on
certain of the maximal class $3$-groups. In fact every maximal class $3$-group
except the Sylow $3$-subgroup of $\mathrm{Sp}_{4}(3)$ has rank $2$. In all of
the examples they discovered, the $3$-group has an abelian subgroup of index
$3$ [8]. In 2019, Parker and Semeraro, using their computational approach to
saturated fusion systems [49] uncovered a saturated fusion system on a rank
$2$ group of order $3^{6}$ which has maximal class and no abelian subgroups of
index $3$. This gave rise to the article [48]. From a different direction,
Clelland and Parker [16] constructed families of saturated fusion systems on a
Sylow $p$-subgroup $T$ of groups of shape $q^{a}:\mathrm{SL}_{2}(q)$ where
$2\leq a\leq p$ and $q=p^{b}$. When $a=2$, $T$ is a Sylow $p$-subgroup of
$\mathrm{SL}_{3}(q)$ and, for $a=3$, $T$ is a Sylow $p$-subgroup of
$\mathrm{Sp}_{4}(q)$. For $a>3$ and for $p\geq 5$, the fusion systems
$\mathcal{F}$ with $O_{p}(\mathcal{F})=1$ discovered in [16] are typically
exotic. If $q=p$, then $T$ has maximal class. This construction therefore
yields infinite families of exotic saturated fusion systems on such maximal
class $p$-groups. In each case, the underlying $p$-group has an abelian
subgroup of index $p$. In a remarkable series of articles, Oliver takes this
property as his starting point and in [42, 20, 44] he, and his co-authors,
determine the reduced saturated fusion systems on $p$-groups with an abelian
subgroup of index $p$. A compilation of their results when applied to
$p$-groups of maximal class is provided in the appendix to this paper. It
turns out that these saturated fusion systems are overwhelmingly exotic. This
perhaps leaves the impression that for odd primes such exotic fusion systems
are not exotic at all. This is possibly be an illusion created by considering
groups which are is some way small. Evidence that they may be exotic after all
comes, for example, from [46] where it is shown that, in certain good
situations, a saturated fusion system determines a locally finite classical
Tits chamber system and so is not exotic. Work of van Beek generalizing the
classification of groups with a weak $BN$-pair has also not revealed any
surprises [55].

Grazian has classified the fusion systems on $p$-groups of rank $3$ for $p\geq
5$ [30]. Unlike in the earlier work of Díaz, Ruiz and Viruel for rank $2$
groups, there is no list of groups to examine such as those given in [8] and
her methods invoke deep results from group theory developed for the
classification of the finite simple groups. The resulting theorem reveals that
saturated fusion systems on such groups are realizable with just one isolated
exotic example on a maximal class $7$-group of order $7^{5}$.

The first step toward a classification of the fusion systems $\mathcal{F}$ on
a class of $p$-groups is the study of the so-called _$\mathcal{F}$ -essential
subgroups_ (see Definition 5.8). In Grazian’s research, a certain special type
of $\mathcal{F}$-essential subgroups played an important role. She named these
subgroups _$\mathcal{F}$ -pearls_. They are defined as follows:

######  Definition 1.1.

Let $p$ be a prime and $\mathcal{F}$ be a saturated fusion system on a
$p$-group $P$. An $\mathcal{F}$-essential subgroup $E$ of $P$ is called an
_$\mathcal{F}$ -pearl_ if it is isomorphic to either the elementary abelian
group of order $p^{2}$, or the extraspecial group of exponent $p$ and order
$p^{3}$ if $p$ is odd or the quaternion of order $8$ if $p=2$. We denote the
set of abelian $\mathcal{F}$-pearls by $\mathcal{P}_{a}(\mathcal{F})$, the set
of extraspecial or quaternion $\mathcal{F}$-pearls by
$\mathcal{P}_{e}(\mathcal{F})$ and we write
$\mathcal{P}(\mathcal{F})=\mathcal{P}_{a}(\mathcal{F})\cup\mathcal{P}_{e}(\mathcal{F})$.

In [29], Grazian develops many of the fundamental properties of saturated
fusion systems which have $\mathcal{F}$-pearls, one of the most basic being
that the underlying group of such saturated fusion systems must have maximal
class. She also presents a fascinating lemma [29, Lemma 3.7] (included here as
Lemma 5.21), that constructs a proper saturated subfusion systems with
$\mathcal{F}$-pearls of any saturated fusion system that has
$\mathcal{F}$-pearls. These subsystems are frequently exotic and if the
$\mathcal{F}$-pearls are non-abelian the systems are simple[5, Definition
I.6.1]. Indeed, the exotic simple saturated fusion system on the group of
order $7^{5}$ discovered in [30] can be constructed from the saturated fusion
system of the monster sporadic simple group at the prime $7$.

It is important to clarify that not all saturated fusion systems on a maximal
class $p$-group contain $\mathcal{F}$-pearls. The determination of saturated
fusion systems $\mathcal{F}$ on $S$ with $O_{p}(\mathcal{F})=1$ and without
$\mathcal{F}$-pearls is a consequence of the main results of this work and is
recorded as Corollary 1.3.

Before we can state our main theorems, we require some further notation.
Assume that $S$ has maximal class and order at least $p^{4}$. Set
$\gamma_{2}(S)=S^{\prime}=[S,S]$ and define
$\gamma_{j}(S)=[\gamma_{j-1}(S),S]$ for $j\geq 3$. If $n\geq 4$, we set
$\gamma_{1}(S)=C_{S}(\gamma_{2}(S)/\gamma_{4}(S))$. Then, as $S$ has maximal
class, $\gamma_{2}(S)/\gamma_{4}(S)$ has order $p^{2}$ and
$|S:\gamma_{1}(S)|=p$. We also have $\gamma_{n-2}(S)$ is the second centre
$Z_{2}(S)$ of $S$, which also has order $p^{2}$. In particular the group
$C_{S}(Z_{2}(S))$, just like $\gamma_{1}(S)$, is a maximal subgroup of $S$.
These subgroups are examples of _$2$ -step centralizers_. If
$\gamma_{1}(S)\neq C_{S}(Z_{2}(S))$, then $S$ is called _exceptional_.
Exceptional maximal class $p$-groups exist for $p\geq 5$ and have order
bounded above by $p^{p+1}$ (see Section 3). The fusion system
$\mathcal{F}_{T}(\mathrm{M})$ of the monster $\mathrm{M}$ on one of its Sylow
$7$-groups $S$ is an example of a saturated fusion system on a maximal class
$p$-group containing an $\mathcal{F}$-pearl and in which the groups
$C_{S}(Z_{2}(S))$ and $\gamma_{1}(S)$ are distinct and
$\mathcal{F}$-essential. An example of an exceptional group of maximal class
is a Sylow $p$-subgroup of the group $\mathrm{G}_{2}(p)$ for $p\geq 5$. The
saturated fusion systems $\mathcal{F}$ on a Sylow $p$-subgroup of the simple
group $\mathrm{G}_{2}(p)$ satisfying $O_{p}(\mathcal{F})=1$ have been
classified by Parker and Semeraro in [47]. Their work uncovers 27 exotic
fusion systems on such $p$-groups when $p=7$. Another example of exceptional
maximal class groups are the $p$-groups of maximal nilpotency class in which
the group $\gamma_{1}(S)$ is extraspecial. Moragues Moncho [41] classified all
saturated fusion systems $\mathcal{F}$ on $p$-groups with an extraspecial
subgroup of index $p$ and $O_{p}(\mathcal{F})=1$. The resulting theorem is
much more uniform than the result concerning $p$-groups with an abelian
subgroup of index $p$. In particular he shows that if the $p$-group considered
has order at least $p^{6}$ then it is either a Sylow $p$-subgroup of
$\mathrm{SL}_{4}(p)$, $\mathrm{SU}_{4}(p)$ or $\mathrm{G}_{2}(p)$ or it is
uniquely determined of order $p^{p-1}$. In the latter case, the fusion systems
discovered are closely related to those found by the second author and Stroth
[50] and are all known.

A consequence of the main results of our research can be presented as follows.
The notation used for sporadic simple groups is consistent with [28].

######  Theorem A.

Suppose that $\mathcal{F}$ is a reduced saturated fusion system on a $p$-group
$S$ of maximal class. Assume that $\gamma_{1}(S)$ is not abelian. Then either
$S$ is not exceptional, $\mathcal{F}$ is simple and exotic and every
$\mathcal{F}$-essential subgroup is an abelian $\mathcal{F}$-pearl or $S$ is
exceptional and one of the following statements holds.

  1. (i)

$p\geq 5$ and $\mathcal{F}=\mathcal{F}_{S}(\mathrm{G}_{2}(p))$;

  2. (ii)

$p=5$, $S$ is isomorphic to a Sylow $5$-subgroup of $\mathrm{G}_{2}(5)$ and
$\mathcal{F}=\mathcal{F}_{S}(G)$ where $G=\operatorname{Ly},\operatorname{HN}$
or $\mathrm{B}$; or

  3. (iii)

$p=7$, $S$ is isomorphic to a Sylow $5$-subgroup of $\mathrm{G}_{2}(7)$ and
either $\mathcal{F}$ is exotic (10 examples) or
$\mathcal{F}=\mathcal{F}_{S}(\operatorname{M})$.

  4. (iv)

$p\geq 11$, $S$ is uniquely determined of order $p^{p-1}$,
$\mathcal{P}(\mathcal{F})=\mathcal{P}_{a}(\mathcal{F})\not=\emptyset$ and, if
$\gamma_{1}(S)$ is $\mathcal{F}$-essential, then
$\mathrm{Out}_{\mathcal{F}}(S)\cong\mathrm{GF}(p)^{\times}\times\mathrm{GF}(p)^{\times}$,
$O^{p^{\prime}}(\mathrm{Out}_{\mathcal{F}}(\gamma_{2}(S)))\cong\mathrm{SL}_{2}(p)$
and $\gamma_{1}(S)/Z(\gamma_{1}(S))$ is the $(p-3)$-dimensional irreducible
$\mathrm{GF}(p)\mathrm{SL}_{2}(p)$-module.

Theorem A is proved by extracting special cases from Theorems B and C.

The maximal class $2$-groups are either dihedral, quaternion or semidihedral
[37, Corollary 3.3.4(iii)] and the fusion systems on such groups are known; in
particular it is easy to demonstrate that the $\mathcal{F}$-essential
subgroups are all $\mathcal{F}$-pearls (see Lemma 6.1). As for $p=3$, the
saturated fusion systems on maximal class $3$-groups are all known due to the
work of Díaz, Ruiz, Viruel and Parker and Semeraro (see Lemma 6.2). Therefore
for our main theorems we focus our attention on the primes $p\geq 5$.

To state our next theorem we require some further notation. Let $S(p)$ be the
unique split extension of an extraspecial group of exponent $p$ and order
$p^{p-2}$ by a cyclic group of order $p$ which has maximal class [41,
Proposition 8.1]. When $p=7$, we have $S(7)$ is isomorphic to the Sylow
$7$-subgroup of $\mathrm{G}_{2}(7)$. The group denoted by
$\mathrm{SmallGroup}(5^{6},661)$ is group number $661$ in the Magma [9] small
group library of groups of order $5^{6}$. We will see that the maximal class
$p$-groups which are exceptional have order at least $p^{6}$ and at most
$p^{p+1}$. So there are no exceptional maximal class $3$-groups. This explains
our assumption that $p\geq 5$ and $|S|\geq p^{6}$ in the next theorem.

######  Theorem B.

Suppose that $p\geq 5$, $S$ is an exceptional maximal class $p$-group of order
at least $p^{6}$ and $\mathcal{F}$ is a saturated fusion system on $S$. Assume
that $\mathcal{F}\neq N_{\mathcal{F}}(S)$. Then one of the following holds.

  1. (i)

$\gamma_{1}(S)$ is extraspecial, and, if $\mathcal{F}\neq
N_{\mathcal{F}}(\gamma_{1}(S))$, then one of the following holds:

    1. (a)

$S$ is isomorphic to a Sylow $p$-subgroup of $\mathrm{G}_{2}(p)$ and either

      1. (enumiii)

$\mathcal{F}=N_{\mathcal{F}}(C_{S}(Z_{2}(S)))$,
$O^{p^{\prime}}(\mathrm{Out}_{\mathcal{F}}(C_{S}(Z_{2}(S))))\cong\mathrm{SL}_{2}(p)$;

      2. (enumiii)

$p=5$, $1\neq O_{p}(\mathcal{F})\leq\gamma_{2}(S)$,
$\mathcal{F}\cong\mathcal{F}_{S}(5^{3}{}^{.}\mathrm{SL}_{3}(5))$;

      3. (enumiii)

$p\geq 5$ and $\mathcal{F}=\mathcal{F}_{S}(\mathrm{G}_{2}(p))$;

      4. (enumiii)

$p=5$ and $\mathcal{F}=\mathcal{F}_{S}(G)$ where
$G=\operatorname{Ly},\operatorname{HN},\mathrm{Aut}(\operatorname{HN})$ or
$\mathrm{B}$; or

      5. (enumiii)

$p=7$ and either $\mathcal{F}$ is exotic (27 examples) or
$\mathcal{F}=\mathcal{F}_{S}(\operatorname{M})$.

    2. (b)

$p\geq 11$, $S\cong S(p)$,
$\mathcal{P}(\mathcal{F})=\mathcal{P}_{a}(\mathcal{F})\not=\emptyset$ and, if
$\gamma_{1}(S)$ is $\mathcal{F}$-essential, then
$\mathrm{Out}_{\mathcal{F}}(S)\cong\mathrm{GF}(p)^{\times}\times\mathrm{GF}(p)^{\times}$,
$O^{p^{\prime}}(\mathrm{Out}_{\mathcal{F}}(\gamma_{2}(S)))\cong\mathrm{SL}_{2}(p)$
and $\gamma_{1}(S)/Z(\gamma_{1}(S))$ is the $(p-3)$-dimensional irreducible
$\mathrm{SL}_{2}(p)$-module.

  2. (ii)

$p=5$, $S=\mathrm{SmallGroup}(5^{6},661)$,
$O_{5}(\mathcal{F})=C_{S}(Z_{2}(S))$ is the unique $\mathcal{F}$-essential
subgroup, $\mathrm{Out}_{\mathcal{F}}(S)$ is cyclic of order $4$,
$\mathrm{Out}_{\mathcal{F}}(C_{S}(Z_{2}(S)))\cong\mathrm{SL}_{2}(5)$ and
$\mathcal{F}$ is unique.

In particular, if $\mathcal{F}\neq N_{\mathcal{F}}(\gamma_{1}(S))$, then
$\mathcal{F}=O^{p}(\mathcal{F})$ and, in addition, $O_{p}(\mathcal{F})=1$ in
all cases other than parts (i)(a)($\alpha$), (i)(a)($\beta$) and part (ii).

Theorem B does not describe $\mathrm{Aut}_{\mathcal{F}}(\gamma_{1}(S))$ when
$\mathcal{F}=N_{\mathcal{F}}(\gamma_{1}(S))$. This additional detail can be
determined as follows. First observe that $|S|\leq p^{p+1}$ as $S$ is
exceptional. Therefore $|\gamma_{1}(S)|=p^{1+2a}$ where $2\leq a\leq(p-1)/2$.
Now $\mathrm{Aut}_{S}(\gamma_{1}(S))$ acts on $\gamma_{1}(S)/Z(S)$ with a
single Jordan block and so we can apply [19] to determine the candidates for
$\mathrm{Out}_{\mathcal{F}}(\gamma_{1}(S))$. For non-exceptional $p$-groups,
we prove the following theorem.

######  Theorem C.

Suppose that $p$ is an odd prime, $S$ is a maximal class $p$-group of order at
least $p^{4}$ and $\mathcal{F}$ is a saturated fusion system on $S$. Assume
that $S$ is not exceptional, $\gamma_{1}(S)$ is not abelian and
$\mathcal{F}\neq N_{\mathcal{F}}(\gamma_{1}(S))$. Then one of the following
holds:

  1. (i)

$\mathcal{E}_{\mathcal{F}}=\mathcal{P}_{a}(\mathcal{F})$,
$|S:\mathrm{hyp}(\mathcal{F})|\leq p$ with $|S:\mathrm{hyp}(\mathcal{F})|=p$
if and only if $|S|=p^{j(p-1)+1}$ for some $j\geq 1$. Furthermore,
$O_{p}(\mathcal{F})=1$ and $O^{p}(\mathcal{F})$ is simple and exotic.

  2. (ii)

$p\geq 5$, $\mathcal{E}_{\mathcal{F}}=\mathcal{P}_{e}(\mathcal{F})$,
$O_{p}(\mathcal{F})=Z(S)$, $|S:\mathrm{hyp}(\mathcal{F})|\leq p$ with
$|S:\mathrm{hyp}(\mathcal{F})|=p$ if and only if $|S|=p^{j(p-1)+2}$ for some
$j\geq 1$. Furthermore, $O^{p}(\mathcal{F}/Z(S))$ is simple and exotic.

  3. (iii)

$p\geq 5$,
$\mathcal{E}_{\mathcal{F}}=\mathcal{P}_{a}(\mathcal{F})\cup\\{\gamma_{1}(S)\\}$,
$O_{p}(\mathcal{F})=1$, $\mathcal{F}\neq O^{p}(\mathcal{F})$ and

    1. (a)

$\mathcal{P}_{a}(\mathcal{F})$ is a single $\mathcal{F}$-class and
$|S|=p^{j(p-1)+1}$ for some $j\geq 2$;

    2. (b)

$\mathrm{Out}_{\mathcal{F}}(\gamma_{1}(S))\cong\mathrm{Sym}(p)$ or
$\mathrm{PGL}_{2}(p)$;

    3. (c)

$Z(\gamma_{1}(S))=\mho^{1}(\gamma_{1}(S))$ has index $p^{p-1}$ in
$\gamma_{1}(S)$, $\gamma_{1}(S)^{\prime}<\Omega_{1}(\gamma_{1}(S))$ has order
$p^{p-2}$ and $\gamma_{2}(S)$ is abelian but not elementary abelian;

    4. (d)

every composition factor of $\mathrm{Aut}_{\mathcal{F}}(\gamma_{1}(S))$ on
$\gamma_{1}(S)$ has order $p$ or $p^{p-2}$ and the composition factors of
order $p$ are centralized by $\mathrm{Aut}_{\mathcal{F}}(\gamma_{1}(S))$;

    5. (e)

$\mathrm{hyp}(\mathcal{F})=P\gamma_{2}(S)$, $O^{p}(\mathcal{F})$ is a
saturated fusion system on $P\gamma_{2}(S)$, and
$\mathrm{Aut}_{O^{p}(\mathcal{F})}(\gamma_{2}(S))\cong\mathrm{Sym}(p)$ or
$\mathrm{PGL}_{2}(p)$.

Suppose that $p$ odd, $r$ a prime with $r^{a}-1\equiv 0\pmod{p}$. In Section
14, we show that an automorphism group $G$ of $G_{0}=\mathrm{PSL}_{p}(r^{ap})$
which projects diagonally into a Sylow $p$-subgroups of $\mathrm{Out}(G_{0})$
between the image of $\mathrm{PGL}_{p}(r^{a})$ and the image of a field
automorphism of order $p$ provides realizable examples of Theorem C (iii)
with, for $S\in\mathrm{Syl}_{p}(G)$,
$\mathrm{Out}_{\mathcal{F}_{S}(G)}(\gamma_{1}(S))\cong\mathrm{Sym}(p)$. By
[49, Theorem 6.2], the subfusion systems generated by the
$\mathcal{F}_{S}(G)$-pearls gives an example of (i) in the case that
$\gamma_{2}(S)$ is abelian. We also remark that in Theorem C (ii) when
$\gamma_{1}(S)/Z(S)$ is abelian, we may apply Oliver’s Theorem [42, Theorem
2.8] to see that $O^{p}(\mathcal{F}/Z(S))$ is simple and exotic. If
$\mathcal{F}$ is a saturated fusion system with
$\mathrm{Out}_{\mathcal{F}}(\gamma_{1}(S))\cong\mathrm{PGL}_{2}(p)$ with
$p\geq 7$ and $\gamma_{1}(S)$ is non-abelian, we may use [44, Theorem 4.5] to
see that $O^{p}(\mathcal{F})$ is exotic.

Theorem C does not specify the structure of a saturated fusion system
$\mathcal{F}=N_{\mathcal{F}}(\gamma_{1}(S))$ and, as we don’t know the
structure of $\gamma_{1}(S)$, it could be more difficult to determine the
precise structure than in the exceptional case. This leads to some
complications in our inductive arguments. Theorems B and C both leave open the
question of what we can say about saturated fusion systems which only have
$\mathcal{F}$-pearls. As an indicator that the structure of $S$ can be more
complicated than the structures in Theorem C we have the following example
which was obtained by computer using the procedures developed in [49] and
implemented in Magma [9]. Note that as $|S|=5^{7}$ in the example, $S$ is not
exceptional.

######  Example 1.2.

Suppose that $S$ is a maximal class $5$-group of order $5^{7}$ and suppose
that $\gamma_{1}(S)$ is not abelian. If $\mathcal{F}$ is a saturated fusion
system on $S$ and $\mathcal{P}(\mathcal{F})$ is non-empty, then $S$ is one of
the six groups

$\mathrm{SmallGroup}(5^{7},1308),\mathrm{SmallGroup}(5^{7},1321),\mathrm{SmallGroup}(5^{7},1360),$
$\mathrm{SmallGroup}(5^{7},1363),\mathrm{SmallGroup}(5^{7},1374),\mathrm{SmallGroup}(5^{7},1384).$

Furthermore, each possibility for $S$ supports a unique (up to isomorphism)
fusion system $\mathcal{F}$ with $\mathcal{F}$-pearls and $\gamma_{1}(S)$ is
not $\mathcal{F}$-essential. For $S=\mathrm{SmallGroup}(5^{7},1308)$,
$\mathcal{F}$ has a unique $\mathcal{F}$-class of extraspecial
$\mathcal{F}$-pearls and the fusion systems on the remaining groups have a
single $\mathcal{F}$-class of abelian $\mathcal{F}$-pearls. If $S$ is one of
$\mathrm{SmallGroup}(5^{7},1360),$ $\mathrm{SmallGroup}(5^{7},1363),$
$\mathrm{SmallGroup}(5^{7},1374),$ or $\mathrm{SmallGroup}(5^{7},1384),$ then
$\gamma_{1}(S)$ has nilpotency class $3$.

All the fusion systems in Example 1.2 with abelian pearls are simple and
exotic by Theorem 5.23. To provide some perspective to this calculation, we
remark that there are 99 maximal class groups of order $5^{7}$. Three of them
have an abelian subgroup of index $5$. Suppose that $S$ is a maximal class
$5$-group with $\gamma_{1}(S)$ non-abelian and let $\mathcal{F}$ be saturated
fusion system on $S$. Assume $\mathcal{F}$ has an $\mathcal{F}$-pearl $P$.
Then Theorem C (i) says that $\mathrm{Aut}(S)$ has a Hall
$5^{\prime}$-subgroup of order $4$. Of the maximal class $5$-groups of order
$5^{7}$, with non-abelian $2$-step centralizer just 14 of them have a Hall
$5^{\prime}$-subgroup of $\mathrm{Aut}(S)$ of order $4$. Of these, 12 have an
elementary abelian subgroup of order $25$ not contained in the $2$-step
centralizer and so have candidates for pearls. We also remark that the fusion
system $\mathcal{F}$ on $S=\mathrm{SmallGroup}(5^{7},1308)$ has a unique class
of extraspecial $\mathcal{F}$-pearls and so this shows that there are examples
in Theorem C (ii) when $p=5$.

The study of maximal class $p$-groups with large automorphism group as in [22]
requires more research to make substantial headway on the problem of
determining all fusion systems which have every essential subgroup a pearl. To
make this statement clearer, Let us fix
$B=\left\\{\left(\begin{smallmatrix}a&0&0\\\ b&a^{-1}&0\\\
c&d&1\end{smallmatrix}\right)\mid a,b,c,d\in\mathrm{GF}(p),a\neq 0\right\\}$.
Let $S$ be maximal class $p$-group of order at least $p^{5}$, and suppose that
$x\in S$ has order $p$ and $P=C_{S}(x)=\langle x,Z(S)\rangle$. Then $N_{S}(P)$
is extraspecial of order $p^{3}$. If $S$ has an automorphism $\phi$ of order
$p-1$ which normalizes $p$ and satisfies
$N_{S}(P)\rtimes\langle\phi\rangle\cong B$. Then there exists a saturated
fusion system $\mathcal{F}$ on $S$ with $P$ an abelian $\mathcal{F}$-pearl
just as in [42, Theorem 2.8]. The existence of an automorphism of order $p-1$
is precisely the condition that Dietrich and Eick use in their work [22].

We now state the corollary as advertised above.

######  Corollary 1.3.

Let $p$ be a prime, $S$ be a $p$-group of maximal nilpotency class and let
$\mathcal{F}$ be a saturated fusion system on $S$ with $O_{p}(\mathcal{F})=1$.
If $\mathcal{P}(\mathcal{F})$ is empty, then $S$ is isomorphic to a Sylow
$p$-subgroups of $\mathrm{G}_{2}(p)$ and either

  1. (i)

$\mathcal{F}=\mathcal{F}_{S}(\mathrm{G}_{2}(p))$;

  2. (ii)

$p=5$ and $\mathcal{F}=\mathcal{F}_{S}(G)$ where
$G=\operatorname{Ly},\operatorname{HN},\mathrm{Aut}(\operatorname{HN})$ or
$\mathrm{B}$;

  3. (iii)

$p=7$, the $\mathcal{F}$-essential subgroups are $C_{S}(Z_{2}(S))$ and
$\gamma_{1}(S)$, $\mathrm{Out}_{\mathcal{F}}(\gamma_{1}(S))\cong 3\times
2^{.}\mathrm{Sym}(7)$,
$\mathrm{Out}_{\mathcal{F}}(C_{S}(Z_{2}(S)))\cong\mathrm{GL}_{2}(7)$ and
$\mathrm{Out}_{\mathcal{F}}(S)\cong\mathbb{Z}_{7}^{\times}\times\mathbb{Z}_{7}^{\times}$
and $\mathcal{F}$ is exotic.

We remark that the exotic saturated fusion system in part (iii) of the
corollary is obtained from the saturated fusion system in the monster
$\operatorname{M}$ by pruning the pearl [49, Lemma 6.5].

The article develops as follows.

We start in Section 2 with some background group theoretical results.
Especially, we explain what it means for a group to have a strongly
$p$-embedded subgroup and present some elementary facts about such groups.

Section 3 commences with two lemmas which detail properties of maximal class
$p$-groups, most of which are drawn from [37]. Of particular importance are
the facts that $\gamma_{1}(S)$ is a regular $p$-group and
$|\Omega_{1}(\gamma_{1}(S))|\leq p^{p}$, with equality if an only if
$|S|=p^{p+1}$. In the second part of Section 3, we start to study the
automorphism group of $S$. As we are interested in the structure of
$\mathrm{Aut}_{\mathcal{F}}(S)$, we specifically study
$p^{\prime}$-automorphisms. Two very important results for our work, which
were known to Juhász [35], are Lemmas 3.9 which describes the action of a
single $p^{\prime}$-automorphism of $S$ on $\gamma_{k}(S)/\gamma_{k+1}(S)$ for
$k\geq 1$, and Lemma 3.10 which says that if some $p^{\prime}$-automorphism of
$S$ centralizes $S/\gamma_{1}(S)$, then $\gamma_{1}(S)$ is either abelian or
$S$ is exceptional and $\gamma_{1}(S)$ is extraspecial. In our work this lemma
has the consequence that most of the time we can assume that
$\mathrm{Out}_{\mathcal{F}}(S)$ is cyclic of order dividing $p-1$. But perhaps
more important than both these results is Lemma 3.13 which applies for $S$
non-exceptional and, for example, controls the size of
$|C_{\gamma_{1}(S)}(\psi)|$ for $\psi$ a $p^{\prime}$-automorphism of $S$. A
particular consequence of Lemma 3.13 is that if $\tau$ is an automorphism of
$S$ and $\tau$ centralizes $\gamma_{k}(S)/\gamma_{k+2}(S)$ then $\tau$ is a
$p$-element. A final important result is Theorem 3.14 which is [35, Theorem
6.2]. This has the consequence that if $\gamma_{w}(S)$ is elementary abelian
and $w\geq 3$, then $\gamma_{w-1}(S)$ has nilpotency class at most $2$. For
completeness, we present a modestly simplified version of his proof.

In Section 4, we gather a collection of results about representations of
groups with cyclic Sylow $p$-subgroup of order $p$. These results are applied
later in the paper in the case when $\gamma_{1}(S)$ is known to be an
$\mathcal{F}$-essential subgroup, in order to obtain the structure of
$\mathrm{Out}_{\mathcal{F}}(\gamma_{1}(S))$. Of particular importance is
Feit’s Theorem (Theorem 4.2) which says that if a group with a cyclic Sylow
$p$-subgroup is not closely related to $\mathrm{PSL}_{2}(p)$, then any
faithful representation is relatively large. This section also contains a
description of the irreducible $\mathrm{GF}(p)\mathrm{SL}_{2}(p)$-modules, and
results which help decompose tensor products of such modules (see Proposition
4.6). These results are useful throughout the proof of our main theorems.

In Section 5 we recall basic definitions and known facts on fusion systems,
referring mostly to [5] (especially for the terminology). We also present a
number of results about saturated fusion systems with $\mathcal{F}$-pearls:
Lemmas 5.17,5.18, 5.19 and 5.21 as well as Theorems 5.20 and 5.23 which are
mostly taken from [29]. The last result of Section 5 is Proposition 5.25 which
allows us in Sections 13 to construct a saturated subfusion system of a
$p$-group with a maximal abelian subgroup.

Section 6 is where the real work starts. Because of the Alperin-Goldschmidt
Theorem, Theorem 5.9, a saturated fusion system $\mathcal{F}$ is entirely
determined by the $\mathcal{F}$-automorphisms of $S$, its
$\mathcal{F}$-essential subgroups and their $\mathcal{F}$-automorphism groups.
Our first significant objective is to determine all the candidates for
$\mathcal{F}$-essential subgroups in a maximal class $p$-group. This results
in the following theorem.

######  Theorem D.

Suppose that $p$ is a prime, $S$ is a $p$-group of maximal class and order at
least $p^{4}$ and $\mathcal{F}$ is a saturated fusion system on $S$. If $E$ is
an $\mathcal{F}$-essential subgroup, then either $E$ is an
$\mathcal{F}$-pearl, or $E=\gamma_{1}(S)$ or $E=C_{S}(Z_{2}(S))$. Furthermore,
if $S$ is exceptional, then
$\mathcal{P}(\mathcal{F})=\mathcal{P}_{a}(\mathcal{F})$.

Thus, in Section 6, we start analyzing the properties of
$\mathcal{F}$-essential subgroups of maximal class $p$-groups. The most
relevant results are Lemma 6.4, that says that an $\mathcal{F}$-essential
subgroup $E$ of $S$ is not an $\mathcal{F}$-pearl if and only if it is
contained in either $\gamma_{1}(S)$ or $C_{S}(Z_{2}(S))$, and Lemma 6.6, in
which we prove that every normal $\mathcal{F}$-essential subgroup of $S$ is a
maximal subgroup of $S$.

In Section 7 we focus our attention on the $\mathcal{F}$-essential subgroups
of $S$ in the case that $S$ is an exceptional group. The goal of this section
is to prove Proposition 7.1, which implies Theorem D in the case in which
$\gamma_{1}(S)$ is extraspecial and gives the first ingredients toward our
proof of Theorem B.

Section 8 considers the case when $S$ is not exceptional and $\gamma_{1}(S)$
is $\mathcal{F}$-essential. Of particular interest is Lemma 8.4 in which we
prove that $\Omega_{1}(\gamma_{1}(S))$ has nilpotency class at most $2$ and
that, if the class is exactly $2$, then
$O^{p^{\prime}}(\mathrm{Out}_{\mathcal{F}}(\gamma_{1}(S)))\cong\mathrm{PSL}_{2}(p)$.
This lemma will be crucial for the proof of Theorem D.

In Section 9, in the case that $S$ is exceptional, Proposition 9.1 states that
if $\gamma_{1}(S)$ is $\mathcal{F}$-essential, then $\gamma_{1}(S)$ is
extraspecial. Thus, in combination with Proposition 7.1, we may assume that,
if $S$ is exceptional, then $\gamma_{1}(S)$ is not $\mathcal{F}$-essential in
our minimal counterexample. The proof of Proposition 9.1 invokes Lemma 8.4,
Proposition 4.6 and a detailed commutator calculation which in the end reaches
contradiction by using a 1902 result due to Burnside (Theorem 2.4).

Sections 10 and 11 contain the series of results that leads to the proof of
Theorem D. Our proof is achieved by contradiction, considering a minimal
counterexample $\mathcal{F}$ to Theorem D (first with respect to the size of
$S$ and then to the number of morphisms). In other words, we consider a
minimal fusion system $\mathcal{F}$ containing an $\mathcal{F}$-essential
subgroup $E$ that is not an $\mathcal{F}$-pearl and is not equal to
$\gamma_{1}(S)$ or $C_{S}(Z_{2}(S))$ (we say that $E$ is a witness). So assume
that $\mathcal{F}$ is such a fusion system. The most important result of
Section 10 is Proposition 10.6 which asserts $O_{p}(\mathcal{F})=1$. The proof
of Proposition 10.6 uses the fact that we know those non-abelian simple groups
which have a $\mathrm{GF}(p)G$-module on which some non-trivial $p$-elements
act with minimal polynomial of degree $2$. The fact that we require originally
goes back to Thompson in unpublished work and we cite Chermak [15] for the
proof. We also show that for $\mathcal{F}$ a minimal counterexample we must
have $p\geq 7$, $p^{7}\leq|S|<p^{2p-4}$ and $\Omega_{1}(\gamma_{1}(S))$ non-
abelian (Lemmas 10.9 and 10.10). With the scene set, in Section 11 we choose a
subgroup $T$ of $\gamma_{1}(S)$ whose automorphism group is not formed by
restrictions of $\mathcal{F}$-automorphisms of $\gamma_{1}(S)$ and that is
maximal first with respect to the order of its normalizer in $S$ and second
with respect to its own order. Note that a witness $E$ is a candidate for $T$,
but the key idea is that $T$ is not necessary $\mathcal{F}$-essential. The
study of the subgroup $T$ is divided in two major cases: the case in which $T$
is $\mathcal{F}$-centric and the case in which $T$ is not
$\mathcal{F}$-centric. We show that either case is impossible and so $T$
cannot exist, proving Theorem D.

With Theorem D proved, Section 12 contains the proof of Theorem B. By this
stage, this is relatively straight forward because of [41], where Moragues
Moncho classified all saturated fusion systems $\mathcal{F}$ on $p$-groups
with an extraspecial subgroup of index $p$ and $O_{p}(\mathcal{F})=1$.

In Section 13 we prove Theorem C. For the proof, we are interested in the case
in which $S$ is not exceptional. Because of Theorem D, once we assume that
$\mathcal{F}\neq N_{\mathcal{F}}(\gamma_{1}(S))$, we know that
$\mathcal{P}(\mathcal{F})$ is non-empty. Thus we choose an $\mathcal{F}$-pearl
$P$. If $\gamma_{1}(S)$ is not $\mathcal{F}$-essential, then we readily obtain
(i) and (ii) of Theorem C. So in Hypothesis 13.6 we assume that
$\gamma_{1}(S)$ is $\mathcal{F}$-essential and non-abelian. In this case we
let $V=\Omega_{1}(Z(\gamma_{1}(S)))$ and $S_{1}=VP$. Hence Proposition 5.25
can be applied to produce a saturated reduced subfusion system of $S_{1}$ in
which $V$ is $\mathcal{F}$-essential. The application of [20] (see the
appendix to this paper for the main result of [20, 44] applied to maximal
class $p$-groups) gives us that $\mathrm{Out}_{\mathcal{F}}(\gamma_{1}(S))$ is
either $\mathrm{PGL}_{2}(p)$ or $\mathrm{Sym}(p)$ and it also dictates the
isomorphism type of $V$ as a
$\mathrm{GF}(p)\mathrm{Out}_{\mathcal{F}}(\gamma_{1}(S))$-module. We then
determine in Lemma 13.11 detailed information about the action of
$\mathrm{Aut}_{\mathcal{F}}(\gamma_{1}(S))$ on $\gamma_{1}(S)$. In particular,
this shows that the chief factors alternate between having order $p$ and being
central and having order $p^{p-2}$. After a few more observations, in Lemma
13.14 we show that $\mathcal{F}\neq O^{p}(\mathcal{F})$; the proof of this
uses detailed knowledge about the submodule structure of exterior squares of
the $(p-2)$-dimensional modules for $\mathrm{PSL}_{2}(p)$ and
$\mathrm{Alt}(p)$. In Lemma 13.15, we show that $P$ must be abelian and so
there are no extraspecial $\mathcal{F}$-pearls and $O_{p}(\mathcal{F})=1$. We
also determine the structure of $\gamma_{1}(S)$. With this information
available, the proof of Theorem C is quickly brought together.

The final section of the paper presents a group which realises examples of
Theorem C and in the appendix we give a description of the reduced fusion
systems on maximal class $p$-groups with $p$ odd and having an abelian
subgroup of index $p$ taken from [20, 44].

### Non-standard notation.

We follow one of [33, 25, 27] for group theoretic notation and we follow
Leedham-Green and McKay [37] for notation and facts surrounding $p$-groups of
maximal class. In particular, for a maximal class $p$-group $R$, $R^{\prime}$
is denoted by $\gamma_{2}(R)$. We apply almost all maps on the right. If $G$
is a groups and $g\in G$ then $c_{g}$ is the conjugation map
$c_{g}:G\rightarrow G$ defined by $xc_{g}=x^{g}=g^{-1}xg$ for $x\in G$. If
$X,Y\leq G$ are groups and $n\in\mathbb{N}$, then $[X,Y;n]$ is defined by
$[X,Y;1]=[X,Y]$ and $[X,Y;n+1]=[[X,Y;n],Y]$. For a field $\mathbb{K}$,
$\mathbb{K}^{\times}$ denotes its multiplicative group.

Our nomenclature for specific groups is for the most part standard; however we
use $\mathrm{Alt}(n)$ and $\mathrm{Sym}(n)$, $n\geq 3$ to respectively denote
the alternating and symmetric groups of degree $n$. For $p$ odd, the
extraspecial groups of order $p^{3}$ are denoted by $p^{1+2}_{+}$ and
$p^{1+2}_{-}$ where the first group has exponent $p$ and the second exponent
$p^{2}$.

##  2\. General group theoretical results

We use the commutator formulae as in [25, Theorem 2.2.1 and Lemma 2.2.2]
regularly and without reference. We also often refer to [25, Sections 5.2 and
5.3] for results concerning coprime action. Here we catalogue less familiar
results.

Let $p$ be a prime. A $p$-group $T$ is _regular_ if, and only if, for all
$x,y\in T$, there exist $g_{1},\dots,g_{t}\in\langle x,y\rangle^{\prime}$ such
that

$(xy)^{p}=x^{p}y^{p}g_{1}^{p}\dots g_{t}^{p}.$

We will need a handful of properties of regular $p$-groups. These properties
say that regular $p$-groups have similar properties to abelian groups with
respect to taking powers. Recall that, for a $p$-group $X$, $\Omega_{1}(X)$ is
the subgroup of $X$ generated by elements of order $p$ and
$\mho^{1}(X)=\langle x^{p}\mid x\in X\rangle$.

######  Lemma 2.1.

Suppose that $P$ is a regular $p$-group and assume that $Q\leq P$. Then

  1. (i)

$Q$ is regular;

  2. (ii)

$\Omega_{1}(P)$ has exponent $p$;

  3. (iii)

$\Omega_{1}(Q)=Q\cap\Omega_{1}(P);$ and

  4. (iv)

$|P/\Omega_{1}(P)|=|\mho^{1}(P)|$.

###### Proof.

Part (i) is obvious from the definition of a regular $p$-group. For (ii) see
[33, Haupsatz III.10.5]. Part (iii) follows from (ii). Part (iv) comes from
[33, Satz III.10.7]. ∎

######  Lemma 2.2.

Assume that $P$ is a regular $p$-group, $T\geq P^{\prime}$ and
$P=T\Omega_{1}(P)$. Then $\mho^{1}(P)=\mho^{1}(T)$.

###### Proof.

Let $x\in P$. Then there is $t\in T$ and $w\in\Omega_{1}(P)$ such that $x=tw$.
By Lemma 2.1 (ii) we have $w^{p}=1$. Thus [37, Lemma 1.2.10 (iii)] together
with $T\geq P^{\prime}\geq\langle t,w\rangle^{\prime}$, imply there exists
$s\in\langle t,w\rangle^{\prime}$ such that

$x^{p}=(tw)^{p}=t^{p}w^{p}s^{p}=t^{p}s^{p}\in\mho^{1}(T).$

Hence $\mho^{1}(P)\leq\mho^{1}(T)\leq\mho^{1}(P)$ and this gives the result. ∎

For a group $P$, we define

$E_{2}(P)=\\{x\in P\mid[x,y,y]=1\text{ for all }y\in P\\}.$

######  Lemma 2.3.

Suppose that $p$ is a prime and $P$ is a $p$-group of nilpotency class $3$.
Then $E_{2}(P)$ is a characteristic subgroup of $P$ which contains $Z_{2}(P)$.

###### Proof.

Obviously $Z_{2}(P)\leq E_{2}(P)$. Assume that $a,b\in E_{2}(P)$ and let $y\in
P$. Then we calculate

$[ab^{-1},y]=[a,y]^{b^{-1}}[b^{-1},y]=[a,y][a,y,b^{-1}][b^{-1},y]\in
C_{P}(y)Z(P)C_{P}(y)=C_{P}(y).$

Hence $E_{2}(P)$ is a subgroup of $P$. Since $E_{2}(P)$ is a characteristic
subset of $P$, $E_{2}(P)$ is a characteristic subgroup of $P$. ∎

######  Theorem 2.4 (Burnside).

Suppose that $p\neq 3$ is a prime and $P$ is a finite $p$-group such that
$P=E_{2}(P)$. Then $P$ has nilpotency class $2$.

###### Proof.

Notice that $[x,y,y]=1$ if and only if $[(y^{-1})^{x},y]=1$ and this is if and
only if $\langle y^{x}\rangle$ and $\langle y\rangle$ commute. Therefore this
theorem dates back to 1902 [13]. ∎

######  Proposition 2.5.

Suppose that $p$ is a prime, $L$ is a group and $P$ is a Sylow $p$-subgroup of
$L$. Let $V$ be a faithful, irreducible $\mathrm{GF}(p)L$-module. If
$[O_{p^{\prime}}(L),P]\neq 1$, then $\dim V\geq p-1$ and $[V,P;p-2]\neq 0$.

###### Proof.

Set $H=O_{p^{\prime}}(L)\langle x\rangle$ where $x\in P$ does not centralize
$O_{p^{\prime}}(H)$. Let $U$ be a non-trivial composition factor for $H$ in
$V$ which is not centralized by $[O_{p^{\prime}}(H),x]$. Then $H$ is
$p$-soluble, $O_{p}(H/C_{H}(U))=1$ and we may apply the Hall-Higman Theorem
[25, Theorem 11.1.1] to $H/C_{H}(U)$ to obtain the minimal polynomial for $x$
on $U$ is $(X-1)^{r}$ with $r\in\\{p,p-1\\}$. In particular, $\dim V\geq r\geq
p-1$ and $[V,P;p-2]\geq[V,x;p-2]=V(x-1)^{(p-2)}\neq 0$. ∎

######  Lemma 2.6.

Suppose that $p$ is a prime, $S$ is a $p$-group, $E,K\leq S$ with $EK$ a
subgroup of $S$. If $N_{K}(E)\leq E$, then $K\leq E$.

###### Proof.

Let $t\in N_{EK}(E)$. Then $t=ek$ for some $e\in E$ and $k\in K$. Thus

$E=E^{t}=E^{ek}=E^{k}$

and so $k\in N_{K}(E)\leq E$. Hence $N_{EK}(E)\leq E$ and this means that
$E=EK\geq K$. ∎

We require the following cohomological type result which is a consequence of a
theorem of Gaschütz.

######  Lemma 2.7.

Suppose that $p$ is a prime, $G$ a group and $V$ a $\mathrm{GF}(p)G$-module.
Let $W=[V,O^{p}(G)]$ and $T\in\mathrm{Syl}_{p}(G)$. Then
$W+C_{V}(T)=W+C_{V}(G)$. In particular, if $C_{V}(G)=0$, then $C_{V}(T)\leq
W$.

###### Proof.

See [39, Lemma C.17]. ∎

###  2.1. Groups with a strongly $p$-embedded subgroup

In this subsection we collects some results about groups with a strongly
$p$-embedded subgroup.

######  Definition 2.8.

Suppose that $p$ is a prime, $H$ is a group and $M$ is a proper subgroup of
$H$ of order divisible by $p$. Then $M$ is strongly $p$-embedded in $H$ if and
only if $M\cap M^{h}$ has order coprime to $p$ for all $h\in H\setminus M$.

It is easy to establish, see [27, Definition 17.11, Proposition 17.11], that
$M$ is strongly $p$-embedded in $H$ if and only if $M$ contains a Sylow
$p$-subgroup $T$ of $H$ and $N_{H}(R)\leq M$ for all $1\neq R\leq T$. In
particular, if $H$ has a strongly $p$-embedded subgroup, then $O_{p}(H)=1$.

######  Lemma 2.9.

Suppose that $p$ is a prime, $H$ is a group and $M$ is strongly $p$-embedded
in $H$. If $K\leq M$ is a subnormal subgroup of $H$, then $K$ is a
$p^{\prime}$-subgroup.

###### Proof.

We may suppose that $N_{H}(K)\not\leq M$. Let $T\in\hbox{\rm Syl}_{p}(K)$.
Then, by the Frattini Argument, $N_{H}(K)=N_{N_{H}(K)}(T)K$ and so
$N_{H}(T)\not\leq M$. Take $h\in N_{H}(T)\setminus M$. Then $T\leq M\cap
M^{h}$ and so, as $M$ is strongly $p$-embedded in $H$, $T=1$. Hence $K$ is a
$p^{\prime}$-subgroup.∎

######  Lemma 2.10.

Suppose that $p$ is a prime and $H$ is a group with a strongly $p$-embedded
subgroup $M$. If $M$ contains an elementary abelian subgroup of order $p^{2}$,
then $O_{p^{\prime}}(H)\leq M$, $M/O_{p^{\prime}}(H)$ is strongly $p$-embedded
in $H/O_{p^{\prime}}(H)$ and $H/O_{p^{\prime}}(H)$ is an almost simple group.

###### Proof.

See [49, Lemma 4.3]. ∎

######  Lemma 2.11.

Suppose that $p$ is a prime, $H$ is a group with a strongly $p$-embedded
subgroup and that $K$ is a normal subgroup of $H$ which commutes with an
element of order $p$. Then $H/K$ has a strongly $p$-embedded subgroup.

###### Proof.

Assume that $M$ is strongly $p$-embedded in $H$. Then $M$ contains a Sylow
$p$-subgroup $T$ of $H$ and $N_{H}(R)\leq M$ for all $1\neq R\leq T$. Since
$K$ centralizes a $p$-element in $H$ and $K$ is normal in $H$ by assumption,
$K\leq M$. Hence $K\leq O_{p^{\prime}}(H)$ by Lemma 2.9. Set
$\overline{H}=H/K$ and assume that $\overline{H}$ and $\overline{M}$ have
order divisible by $p$. Let $\overline{R}$ be a non-trivial $p$-subgroup of
$\overline{T}$ with $R\leq T$. Then, by coprime action,
$N_{\overline{H}}(\overline{R})=\overline{N_{H}(R)}\leq\overline{M}$ and so we
conclude $\overline{H}$ has a strongly $p$-embedded subgroup as claimed. ∎

######  Proposition 2.12.

Suppose that $p$ is a prime, $X$ is a $\mathcal{K}$-group, $K=F^{*}(X)$ is
simple and $T\in\mathrm{Syl}_{p}(X)$. If $T$ contains an elementary abelian
subgroup of order $p^{2}$ and $X$ has a strongly $p$-embedded subgroup, then
$K$ has a strongly $p$-embedded subgroup and $p$ and $K$ are as follows.

  1. (i)

$p$ is arbitrary, $a\geq 1$ and $K\cong\mathrm{PSL}_{2}(p^{a+1})$, or
$\mathrm{PSU}_{3}(p^{a})$, ${}^{2}\mathrm{B}_{2}(2^{2a+1})$ $(p=2)$ or
${}^{2}{\rm G}_{2}(3^{2a+1})$ $(p=3)$ and $X/K$ is a $p^{\prime}$-group.

  2. (ii)

$p>3$, $K\cong\mathrm{Alt}(2p)$ and $|X/K|\leq 2$.

  3. (iii)

$p=3$, $K\cong\mathrm{PSL}_{2}(8)$ and $X\cong\mathrm{PSL}_{2}(8){:}3$.

  4. (iv)

$p=3$, $K\cong\mathrm{PSL}_{3}(4)$ and $X/K$ is a $2$-group.

  5. (v)

$p=3$ and $X=K\cong\mathrm{M}_{11}$.

  6. (vi)

$p=5$ and $K\cong{}^{2}\mathrm{B}_{2}(32)$ and
$X\cong{}^{2}\mathrm{B}_{2}(32){:}5$.

  7. (vii)

$p=5$, $K\cong{}^{2}\mathrm{F}_{4}(2)^{\prime}$ and $|X/K|\leq 2$.

  8. (viii)

$p=5$, $K\cong\mathrm{McL}$ and $|X/K|\leq 2$.

  9. (ix)

$p=5$, $K\cong\mathrm{Fi}_{22}$ and $|X/K|\leq 2$.

  10. (x)

$p=11$ and $X=K\cong\mathrm{J}_{4}$.

###### Proof.

See [28, 7.6.1] and [26, 24.1] to covers the cases listed in (iii) and (iv).∎

##  3\. Maximal class $p$-groups

Throughout this section, $S$ represents a $p$-group of maximal class of order
$p^{n}$ with $n\geq 3$. This means that $S$ has nilpotency class $n-1$. The
maximal class $2$-groups are the dihedral groups, generalized quaternion
groups and semidihedral groups and so everything about these groups is easy to
calculate. Thus for this section we concentrate on the case when $p$ is odd,
even though many of the results are obviously true when $p=2$. We set
$Z_{1}(S)=Z(S)$ and, for $2\leq j\leq n-1$, $Z_{j}(S)$ is the complete pre-
image of $Z(S/Z_{j-1}(S))$. So the $Z_{j}(S)$ are the terms of the _upper
central series_ of $S$. Similarly, we put $\gamma_{2}(S)=S^{\prime}=[S,S]$ and
define $\gamma_{j}(S)=[\gamma_{j-1}(S),S]$ for $j\geq 3$. These are the
members of the _lower central series_ of $S$. Notice that $\gamma_{j}(S)=1$
for $j\geq n$. As $S$ has maximal class, $Z_{j}(S)=\gamma_{n-j}(S)$ for $1\leq
j\leq n-2$ and this subgroup has order $p^{j}$ and index $p^{n-j}$ in $S$. The
_$2$ -step centralizers _ in $S$ are the subgroups

$C_{S}(\gamma_{j}(S)/\gamma_{j+2}(S))$

where $2\leq j\leq n-2$. As usual, we define

$\gamma_{1}(S)=C_{S}(\gamma_{2}(S)/\gamma_{4}(S)).$

Notice that, if $|S|\geq p^{4}$, then
$|S:\gamma_{1}(S)|=|\gamma_{1}(S):\gamma_{2}(S)|=p$ and all the $2$-step
centralizers in $S$ are maximal subgroups of $S$. A group with more than one
$2$-step centralizer is called an _exceptional group_.

By [37, Lemma 1.1.25 (i)], we always have
$[\gamma_{i}(S),\gamma_{j}(S)]\leq\gamma_{i+j}(S)$. Hence $\gamma_{t}(S)$ is
abelian for all $t\geq n/2$. The _degree of commutativity_ of $S$, is the
greatest integer $c$ such that
$[\gamma_{i}(S),\gamma_{j}(S)]\leq\gamma_{i+j+c}(S)$ for all $1\leq i\leq
j\leq n$. We say that $S$ has a _positive degree of commutativity_ if and only
if $c$ is positive.

The maximal class $2$-groups are the dihedral groups, generalized quaternion
groups and semidihedral groups and so everything about these groups is known.
Thus for this section we concentrate on the case when $p$ is odd even though
many of the results are obviously true when $p=2$. The maximal class
$p$-groups of order $p^{3}$ are extraspecial and we assume that the reader is
familiar with their structure.

######  Lemma 3.1.

Suppose that $S$ is a maximal class $p$-group of order $p^{n}$ with $n\geq 4$.

  1. (i)

If $N$ is a proper normal subgroup of $S$, then either $N=\gamma_{j}(S)$ for
some $j\geq 2$ or $N$ is a maximal subgroup of $S$. Furthermore,
$S/\gamma_{j}(S)$ has maximal class for all $j\geq 2$.

  2. (ii)

$\gamma_{1}(S)$ is a regular $p$-group.

  3. (iii)

If $n>{p+1}$, then $\Omega_{1}(\gamma_{1}(S))=\gamma_{n-(p-1)}(S)$ has
exponent $p$ and order $p^{p-1}$ and
$\mho^{1}(\gamma_{i}(S))=\gamma_{i+(p-1)}(S)$ for all $1\leq i\leq n-(p-1)$.

  4. (iv)

$|\Omega_{1}(\gamma_{1}(S))|\leq p^{p}$ and if equality holds then
$\Omega_{1}(\gamma_{1}(S))=\gamma_{1}(S)$ and $|S|=p^{p+1}$.

  5. (v)

If $n>p+1$, $A\leq\gamma_{1}(S)$ and $B$ is normal in $A$ with $A/B$
elementary abelian, then $|A/B|\leq p^{p-1}$.

  6. (vi)

If $n\leq{p+1}$, then $\gamma_{2}(S)$ and $S/Z(S)$ have exponent $p$.

  7. (vii)

There exists a unique maximal abelian normal subgroup $\gamma_{w}(S)$ of $S$
and either $\Omega_{1}(\gamma_{1}(S))\leq\gamma_{w}(S)$ or $\gamma_{w}(S)$ is
elementary abelian.

###### Proof.

The first part of (i) is [37, Proposition 3.1.2] while the second is obvious
by definition.

For part (ii), if $n\leq p+1$, then $|\gamma_{1}(S)|\leq p^{p}$ and [37, Lemma
1.2.11] yields $\gamma_{1}(S)$ is regular. If $n>p+1$, [37, Corollary 3.3.4
(i)] gives the same result.

For statement (iii) we first note that $\gamma_{1}(S)$ is regular by (ii) and
so $\Omega_{1}(\gamma_{1}(S)$ has exponent $p$ by Lemma 2.1 (ii). The result
now follows from [37, Corollary 3.3.6(i)].

Part (iv) is a consequence of (iii).

Part (v) is included in [37, Exercise 3.3 (3)] and part (vi) is [37,
Proposition 3.3.2].

Part (vii) follows from (i) as $n\geq 4$.

∎

######  Lemma 3.2.

Suppose that $S$ is a maximal class $p$-group of order $p^{n}$ with $n\geq 4$
and $M$ is a maximal subgroup of $S$.

  1. (i)

If $M\not\in\\{\gamma_{1}(S),C_{S}(Z_{2}(S))\\}$, then $M$ has maximal class
and $\gamma_{i}(M)=\gamma_{i+1}(S)$ for $i=1,\dots,n-1$.

  2. (ii)

$M$ is a $2$-step centralizer in $S$ if and only if
$M\in\\{\gamma_{1}(S),C_{S}(Z_{2}(S))\\}$.

  3. (iii)

$S$ is exceptional if and only if $\gamma_{1}(S)\neq C_{S}(Z_{2}(S))$.

  4. (iv)

The degree of commutativity of $S$ is positive if and only if $S$ is not
exceptional.

  5. (v)

If $n$ is odd, or $n=4$ or $n>p+1$, then $S$ is not exceptional.

  6. (vi)

Assume that $|S|\geq p^{5}$. Then $S/Z(S)$ is not exceptional.

  7. (vii)

$\gamma_{1}(S)=C_{S}(\gamma_{j}(S)/\gamma_{j+2}(S))$ for all $2\leq j\leq
n-3$.

###### Proof.

For statement (i) see [56, Lemma (1.2)].

For (ii), we know that $\gamma_{1}(S)$ and $C_{S}(Z_{2}(S))$ are $2$-step
centralizers by definition. That they are the only $2$-step centralizers
follows from (i).

Part (iii) follows from (i) and the definition of an exceptional group.

Part (iv) is [37, Corollary 3.2.7].

Part (v) follows from [37, Theorems 3.2.11 and 3.3.5] for $|S|\geq p^{5}$. If
$|S|=p^{4}$ then $\gamma_{1}(S)={C}_{S}({Z}_{2}(S))$ is abelian and is the
unique $2$-step centralizer.

For part (vi). If $n$ is even, then $|S/Z(S)|=p^{n-1}\geq p^{5}$ is not
exceptional by (v). Whereas, if $n$ is odd, then $S$ is not exceptional by (v)
and $\gamma_{1}(S)$ is the unique $2$-step centralizer in $S$ by (ii). Since
the preimage of a $2$-step centralizer in $S/Z(S)$, is a $2$-step centralizer
in $S$, we deduce that $S/Z(S)$ is not exceptional.

Finally (vii) follows from (vi).

∎

######  Lemma 3.3.

The following hold:

  1. (i)

Suppose that $t\in S$ and $t\not\in\gamma_{1}(S)\cup C_{S}(Z_{2}(S))$. Then
$C_{S}(t)=\langle t,Z(S)\rangle$ has order $p^{2}$ and
$C_{S}(t)\cap\gamma_{1}(S)=C_{S}(t)\cap C_{S}(Z_{2}(S))=Z(S)$.

  2. (ii)

If $T$ is a $p$-group and there exists $t\in T$ with $|C_{T}(t)|=p^{2}$, then
$T$ has maximal class.

###### Proof.

Suppose that $t\not\in\gamma_{1}(S)\cup C_{S}(Z_{2}(S))$. Then $t$ is not
contained in any $2$-step centralizer by Lemma 3.2 (ii).

Assume for a contradiction that $|C_{S}(t)|\geq p^{3}$. Then there exists
$y\in(C_{S}(t)\cap\gamma_{1}(S))\setminus Z(S)$ so that
$y\in\gamma_{j}(S)\setminus\gamma_{j+1}(S)$ with $\gamma_{j+1}(S)\geq Z(S)$.
Now $t$ centralizes

$\gamma_{j}(S)/\gamma_{j+2}(S)=\langle\gamma_{j+1}(S)/\gamma_{j+2}(S),y\gamma_{j+2}(S)\rangle,$

contrary to $t$ not being in a $2$-step centralizer. Since $\gamma_{1}(S)$ and
$C_{S}(Z_{2}(S))$ have index $p$ in $S$ and $t\not\in\gamma_{1}(S)\cup
C_{S}(Z_{2}(S))$, $C_{S}(t)\cap\gamma_{1}(S)=C_{S}(t)\cap
C_{S}(Z_{2}(S))=Z(S)$. This proves (i).

Part (ii) is [33, Satz III.14.23]. ∎

The next lemma provides a weak upper bound for the number of commutators by
$\gamma_{1}(S)$ required to annihilate $\Omega_{1}(\gamma_{1}(S))$.

######  Lemma 3.4.

Suppose that $S$ is a maximal class $p$-group of order at least $p^{4}$. Then
$[\Omega_{1}(\gamma_{1}(S)),\gamma_{1}(S);\frac{p+1}{2}]=1$. Furthermore, if
$[\Omega_{1}(\gamma_{1}(S)),\gamma_{1}(S);\frac{p-1}{2}]\neq 1$, then $S$ is
exceptional.

###### Proof.

If $|S|=p^{4}$, then $\gamma_{1}(S)$ is abelian and so the statement holds in
this case. Suppose $|S|\geq p^{5}$. Lemmas 3.1 (iv) and 3.2 (vi) give that
$|\Omega_{1}(\gamma_{1}(S))|\leq p^{p}$ and $S/Z(S)$ is not exceptional.
Hence, the fact that $\gamma_{1}(S)/Z(S)$ is the unique $2$-step centralizer
of $S/{Z}(S)$ implies
$[\Omega_{1}(\gamma_{1}(S)),\gamma_{1}(S);\frac{p-1}{2}]\leq Z(S)$. This
proves the first bound.

Suppose now that $[\Omega_{1}(\gamma_{1}(S)),\gamma_{1}(S);\frac{p-1}{2}]\neq
1$. We first show that

$[\Omega_{1}(\gamma_{1}(S)),\gamma_{1}(S);\frac{p-1}{2}-1]\leq Z_{2}(S).$

This is clear if $|\Omega_{1}(\gamma_{1}(S))|\leq p^{p-1}$, since
$\gamma_{1}(S)$ is the unique $2$-step centralizer of $S/Z(S)$. Assume
$|\Omega_{1}(\gamma_{1}(S))|=p^{p}$. Then
$\Omega_{1}(\gamma_{1}(S))=\gamma_{1}(S)$ by Lemma 3.1(iv) and we know that
$\gamma_{1}(S)/\gamma_{4}(S)$ is abelian. Hence, also in this case we have
$[\Omega(\gamma_{1}(S)),\gamma_{1}(S);\frac{p-1}{2}-1]\leq Z_{2}(S)$.
Therefore we get
$1\neq[\Omega_{1}(\gamma_{1}(S)),\gamma_{1}(S);\frac{p-1}{2}]\leq[Z_{2}(S),\gamma_{1}(S)]$.
This means that $Z_{2}(S)\not\leq{Z}(\gamma_{1}(S))$ and so $S$ is exceptional
by Lemma 3.2(iii). ∎

######  Lemma 3.5.

Suppose that $S$ is a maximal class $p$-group of order at least $p^{4}$. If
$T\leq S$ and $T\not\subseteq\gamma_{1}(S)\cup C_{S}(Z_{2}(S))$, then $T$ has
maximal class.

###### Proof.

Suppose that $S$ is a counterexample of minimal order and let $T\leq S$ with
$T\not\subseteq\gamma_{1}(S)\cup C_{S}(Z_{2}(S))$ be a subgroup that does not
have maximal class. In particular, $T$ is not abelian of order $p$ or $p^{2}$,
and $T\neq S$. So we have $|T|\geq p^{3}$. Let $M$ be a maximal subgroup of
$S$ which contains $T$. Then $M\not\subseteq\gamma_{1}(S)\cup
C_{S}({Z}_{2}(S))$ and so $M$ has maximal class by Lemma 3.2 (i). Therefore
$T<M$ and $|M|\geq p^{4}$. By Lemma 3.2 (i), $\gamma_{1}(M)=\gamma_{2}(S)$ and
$Z_{2}(M)=Z_{2}(S)$. In particular, $\gamma_{1}(M)$ centralizes $Z_{2}(S)$ and
so $\gamma_{1}(M)=C_{M}(Z_{2}(M))$ and $T\not\subseteq\gamma_{1}(M)\cup
C_{M}(Z_{2}(M))$. Using the minimality of $S$, we conclude that $T$ has
maximal class, a contradiction. ∎

######  Lemma 3.6.

Suppose that $S$ is a maximal class $p$-group of order at least $p^{4}$. If
$T\subseteq\gamma_{1}(S)\cup C_{S}(Z_{2}(S))$ and
$N_{S}(T)\not\subseteq\gamma_{1}(S)\cup C_{S}(Z_{2}(S))$, then $T$ is normal
in $S$. Furthermore, if $Z(S)\leq W\subseteq\gamma_{1}(S)\cup C_{S}(Z_{2}(S))$
and $N_{S}(W)\not\leq\gamma_{1}(S)$, then $W$ is normal in $S$.

###### Proof.

Suppose that $S$ is a counterexample of minimal order and let
$T\subseteq\gamma_{1}(S)\cup C_{S}(Z_{2}(S))$ with
$N_{S}(T)\not\subseteq\gamma_{1}(S)\cup C_{S}(Z_{2}(S))$ be such that
$N_{S}(T)\neq S$. Set $N=N_{S}(T)$ and note $S>N\geq Z(S)$ and $T\neq Z(S)$.
By Lemma 3.5 and the hypothesis $N\not\subseteq\gamma_{1}(S)\cup
C_{S}(Z_{2}(S))$, $N$ has maximal class and $N$ is contained in some maximal
subgroup $M$ of $S$ which also has maximal class. Notice that

$M\cap C_{S}(Z_{2}(S))=M\cap\gamma_{1}(S)=\gamma_{2}(S).$

Thus $T\leq\gamma_{2}(S).$

If $N$ is abelian, then $|N|=p^{2}$ and so $N=TZ(S)\subseteq\gamma_{1}(S)\cup
C_{S}(Z_{2}(S))$, a contradiction. Hence $N$ is non-abelian. Since $N$ has
maximal class and is non-abelian, $Z(S)=Z(N)$ has order $p$. Therefore, as $T$
is normal in $N$, $Z(S)=Z(N)\leq T$. Because $T\neq Z(S)$, $T>Z(S)$.

If $|S/Z(S)|=p^{3}$, then $\gamma_{1}(S)=C_{S}(Z_{2}(S)$),
$T\leq\gamma_{1}(S)$ and $[\gamma_{1}(S)\colon T]\leq p$. Thus $T$ is normal
in $\gamma_{1}(S)$, which is impossible as $N\nleq\gamma_{1}(S)$ and $N\neq
S$. Hence $|S/Z(S)|\geq p^{4}$. Furthermore, $S/Z(S)$ has maximal class and

$T/Z(S)\leq\gamma_{1}(S)/Z(S)=\gamma_{1}(S/Z(S))=C_{S/Z(S)}(Z_{2}(S/Z(S))),$

where the last equality uses Lemma 3.2 (iii) and (vi). Since
$N_{S/Z(S)}(T/Z(S))\geq N/Z(S)$, induction yields $T/Z(S)$ is normal in
$S/Z(S)$ and so $T$ is normal in $S$, which is a contradiction. This proves
the main claim.

Finally, suppose that $Z(S)\leq W\subseteq\gamma_{1}(S)\cup C_{S}(Z_{2}(S))$
and $N_{S}(W)\not\leq\gamma_{1}(S)$. Then

$N_{S/Z(S)}(W/Z(S))=N_{S}(W)/Z(S)\not\leq\gamma_{1}(S/Z(S))=C_{S/Z(S)}(Z_{2}(S/Z(S))),$

where as before the last equality follows from Lemma 3.2 (iii) and (vi).
Hence, by what we have just proved, $W/Z(S)$ is normal in $S/Z(S)$ which means
$W$ is normal in $S$. ∎

We close this subsection by determining some of the finite simple groups which
have a maximal class Sylow $p$-subgroup.

######  Lemma 3.7.

Suppose that $p\geq 5$ is a prime, $G$ is a finite simple group and
$S\in\mathrm{Syl}_{p}(G)$. Assume that $S$ has maximal class and has no
abelian subgroup of index $p$. Then $S$ is isomorphic to a Sylow $p$-subgroup
of $\mathrm{G}_{2}(p)$, and either $G\cong\mathrm{G}_{2}(p)$ or $p=5$ and
$G\cong\operatorname{Ly},\operatorname{HN}$ or $\mathrm{B}$ or $p=7$ and
$G\cong\operatorname{M}$.

###### Proof.

If $G$ is an alternating group of degree $d$, then, as $S$ has no abelian
subgroup of index $p$, $d\geq p^{3}$. But then $S$ contains $p^{p}$ commuting
$p$-cycles. This contradicts Lemma 3.1 (iv).

Suppose $G$ is a Lie type group in characteristic $p$. We use the fact that
$|S/\gamma_{2}(S)|=p^{2}$. Using [28, Theorem 3.3.1] we see that $G$ has
untwisted rank at most $2$ and that it is defined over $\mathrm{GF}(p)$ if it
has rank 2. Since $S$ has no abelian subgroup of index $p$, $G$ is not
$\mathrm{PSL}_{2}(p^{2})$, $\mathrm{PSU}_{3}(p)$, $\mathrm{PSL}_{3}(p)$,
$\mathrm{PSp}_{4}(p)$. This leaves $\mathrm{G}_{2}(p)$. Hence the result holds
for Lie type groups in characteristic $p$.

Suppose that $G$ is a Lie type group in characteristic $r$ with $r\neq p$. We
use [28, Theorem 4.10.2]. Thus we can write $S=P_{T}P_{W}$ where $P_{T}$ is an
abelian normal subgroup of $S$ and $P_{W}$ is a complement to $P_{T}$. Since
$\gamma_{2}(S)=[S,S]=[P_{T}P_{W},P_{T}P_{W}]=[P_{T},P_{W}]P_{W}^{\prime}$,
$|S/\gamma_{2}(S)|=p^{2}$ implies that $P_{W}/P_{W}^{\prime}$ has order $p$.
But then, $P_{W}$ has order $p$ and $S$ has an abelian subgroup of index $p$,
a contradiction.

Finally, suppose that $G$ is a sporadic simple group. We deploy the tables in
[28, Tables 5.3]. The sporadic groups listed in the conclusion of our lemma
have Sylow $p$-subgroups isomorphic to those of $\mathrm{G}_{2}(5)$ in the
first three cases and to those of $\mathrm{G}_{2}(7)$ in the last case. Since
$p\geq 5$, we only need to think about the Sylow $5$-subgroup of the monster
$\operatorname{M}$. This group has order $5^{9}$. Using [28, Table 5.3z], the
monster has a $5$-local subgroup $5^{1+6}.((4*2^{.}\mathrm{J}_{2}).2)$. Thus
$S/Z(S)$ is of maximal class and has a normal subgroup of $5$ rank $6$, as
$|S/Z(S)|=5^{8}>5^{7}$, this contradicts Lemma 3.1(v). ∎

We give a list of the realizable fusion systems on maximal class $p$-groups
which have an abelian subgroup of index $p$ in the appendix Table 3. This
table is extracted from [20, Table 2.2].

###  3.1. Automorphisms of $p^{\prime}$-order

We now present some conclusions about the automorphism group of $p$-groups $S$
of maximal class of order $p^{n}$ with $n\geq 4$. Observing that
$|S/\gamma_{2}(S)|=p^{2}$ and $\gamma_{1}(S)$ is a characteristic subgroup of
$S$, yields the first result which is well-known.

######  Theorem 3.8.

Suppose that $S$ is a $p$-group of maximal class and order at least $p^{4}$.
Then $\mathrm{Aut}(S)/O_{p}(\mathrm{Aut}(S))$ is isomorphic to a subgroup of
the diagonal matrices in $\mathrm{GL}_{2}(p)$. In particular, if $H$ is a
subgroup of $\mathrm{Aut}(S)$ and $|H|$ is coprime to $p$, then $H$ is abelian
and is isomorphic to a subgroup of the diagonal matrices in
$\mathrm{GL}_{2}(p)$. ∎

One consequence of Theorem 3.8 is that if $\varphi\in\mathrm{Aut}(S)$ has
order coprime to $p$, then by Maschke’s Theorem there exists a maximal
subgroup $M$ of $S$ with $M\neq\gamma_{1}(S)$ such that $M\varphi=M$. Choose
$x\in M\setminus\gamma_{1}(S)$ and
$s_{1}\in\gamma_{1}(S)\setminus\gamma_{2}(S)$ and define

$s_{i}=[x,s_{i-1}]~{}\text{ for every }~{}2\leq i\leq n-2~{}\text{ and }$
$s_{n-1}=\begin{cases}[x,s_{n-2}]\quad\text{ if }M={C}_{S}({Z}_{2}(S))\\\
[s_{1},s_{n-2}]\quad\text{otherwise.}\end{cases}$

The choice of $x$ and $s_{1}$ is plainly not unique, and it is even possible
that $[s_{1},s_{2}]\in\gamma_{j}(S)$ and for a different choice of $x$ and
$s_{1}$ we have $[s_{1},s_{2}]\in\gamma_{k}(S)$ with $k<j$. In particular,
there are examples of maximal class groups $S$ and a choice of $x$ and $s_{1}$
such that $[s_{1},s_{2}]=1$ and $\gamma_{1}(S)$ is non-abelian.

######  Lemma 3.9.

Suppose that $S$ is a $p$-group of maximal class and order at least $p^{4}$.
Let $\varphi\in\mathrm{Aut}(S)$ be an automorphism of order coprime to $p$ and
let $x\in S\setminus\gamma_{1}(S)$ be such that $\varphi$ leaves invariant
$\langle x\rangle\gamma_{2}(S)$. If $a,b\in\mathrm{GF}(p)$ are such that
$x\varphi\equiv x^{a}\mod\gamma_{2}(S)$ and $s_{1}\varphi\equiv
s_{1}^{b}\mod\gamma_{2}(S)$ then

$\begin{split}s_{i}\varphi&\equiv s_{i}^{a^{i-1}b}\mod\gamma_{i+1}(S)~{}\text{
for every }~{}1\leq i\leq n-2~{}\text{ and }\\\
s_{n-1}\varphi&=\begin{cases}s_{n-1}^{a^{n-2}b}&\text{ if }S\text{ is not
exceptional}\\\ s_{n-1}^{a^{n-3}b^{2}}&\text{ if }S\text{ is
exceptional.}\end{cases}\end{split}$

###### Proof.

See the proof of [29, Theorem 2.19]. In [29, Theorem 2.19] a stronger result
is stated under the assumption that $\gamma_{1}(S)$ is not abelian or
extraspecial; however the description of the action of $\varphi$ does not use
this additional hypothesis. ∎

######  Lemma 3.10.

Suppose that $S$ is a $p$-group of maximal class and order at least $p^{4}$.
Assume that $\gamma_{1}(S)$ is not abelian nor extraspecial. Then every non-
trivial automorphism of $S$ of order coprime to $p$ acts faithfully on
$S/\gamma_{1}(S)$.

###### Proof.

Suppose $\alpha\in\mathrm{Aut}(S)$ has order coprime to $p$ and centralizes
$S/\gamma_{1}(S)$. We show that $\alpha$ is the trivial automorphism. Aiming
for a contradiction, suppose that $\alpha$ is non-trivial. Let
$b\in\mathrm{GF}(p)$ be such that $s_{1}\alpha\equiv
s_{1}^{b}\mod\gamma_{2}(S)$. Because $\alpha$ is non-trivial, $b\neq 1$ as
$\alpha$ acts faithfully on $S/\gamma_{2}(S)$ by [25, Theorem 5.1.4].
Employing Lemma 3.9 with $a=1$, we get

$\begin{split}s_{i}\alpha&\equiv s_{i}^{b}\mod\gamma_{i+1}(S)~{}\text{ for
every }~{}1\leq i\leq n-2~{}\text{ and }\\\
s_{n-1}\alpha&=\begin{cases}s_{n-1}^{b}&\text{ if }S\text{ is not
exceptional}\\\ s_{n-1}^{b^{2}}&\text{ if }S\text{ is
exceptional}\end{cases}\end{split}$

Since $\gamma_{1}(S)$ is not abelian, we have
${Z}(\gamma_{1}(S))=\gamma_{k+1}(S)$ for some $2\leq k\leq n-2$. Then
$\gamma_{k}(S)$ is abelian and is non-central. Choose $j$ maximal such that
$[s_{k},s_{j}]\neq 1$. As $[s_{j},s_{k}]\neq 1$,
$[s_{j},s_{k}]\in\gamma_{i}(S)\backslash\gamma_{i+1}(S)$ for some $k+1\leq
i\leq n-1$. Note that $[s_{j},s_{k}]\in{Z}(\gamma_{1}(S))$ commutes with both
$s_{j}$ and $s_{k}$. Also, $s_{j}\alpha=s_{j}^{b}z$ for some
$z\in\gamma_{j+1}(S)$, that commutes with $s_{k}$ by the maximal choice of
$j$, and $s_{k}\alpha=s_{k}^{b}u$, for some
$u\in\gamma_{k+1}(S)={Z}(\gamma_{1}(S))$. Therefore using the commutator
formulae we get

$[s_{j},s_{k}]\alpha=[s_{j}\alpha,s_{k}\alpha]=[s_{j}^{b}z,s_{k}^{b}u]=[s_{j}^{b},s_{k}^{b}]=[s_{j},s_{k}]^{b^{2}}.$

Suppose $S$ is not exceptional. Then

$[s_{j},s_{k}]^{b^{2}}\equiv[s_{j},s_{k}]^{b}\mod\gamma_{i+1}(S)$

and so $b=1$, which is a contradiction.

Suppose $S$ is exceptional. Then $\overline{S}=S/{Z}(S)$ is not exceptional by
Lemma 3.2(vi) and $\alpha$ is a non-trivial automorphism of $\overline{S}$ of
order coprime to $p$ centralizing $\overline{S}/\gamma_{1}(\overline{S})$.
Since $S/Z(S)$ is not exceptional, using what we proved above we obtain
$\gamma_{1}(\overline{S})=\gamma_{1}(S)/{Z}(S)$ is abelian. Hence
$\gamma_{1}(S)$ is extraspecial, contradicting the hypothesis. ∎

######  Corollary 3.11.

Suppose that $S$ is a $p$-group of maximal class and order at least $p^{4}$.
Assume that $\gamma_{1}(S)$ is not abelian nor extraspecial. If $H$ is a
subgroup of $\mathrm{Aut}(S)$ of order coprime to $p$, then $H$ is cyclic of
order $m$ dividing $(p-1)$ and $H$ acts faithfully on $S/\gamma_{1}(S)$.
Furthermore, $|\mathrm{Aut}(S)|=p^{a}m$ for some natural number $a$.

###### Proof.

We know that $\gamma_{1}(S)$ is a characteristic subgroup of $S$. Hence there
is a homomorphism
$\theta:\mathrm{Aut}(S)\rightarrow\mathrm{Aut}(S/\gamma_{1}(S))$ and the
latter group is cyclic of order $p-1$. By Lemma 3.10, $\ker\theta$ is a
$p$-group. This proves the claim. ∎

######  Lemma 3.12.

Suppose that $S$ is a $p$-group of maximal class and order at least $p^{4}$.
If $S$ is exceptional and $\alpha\in\mathrm{Aut}(S)$ is an involution which
inverts $S/\gamma_{1}(S)$, then $\alpha$ inverts $Z(S)$.

###### Proof.

Since $\alpha$ is an involution which inverts $S/\gamma_{1}(S)$,
$(x\gamma_{2}(S))\alpha=x^{-1}\gamma_{2}(S)$ and
$(s_{1}\gamma_{2}(S))\alpha=s_{1}^{b}\gamma_{2}(S)$ where $b=\pm 1$. By Lemma
3.9, $\alpha$ acts on $Z(S)$ as

$(-1)^{n-3}b^{2}=(-1)^{n-3}=-1.$

Hence $\alpha$ inverts $Z(S)$. ∎

######  Lemma 3.13.

Suppose that $S$ is a $p$-group of maximal class and order at least $p^{4}$
that is not exceptional. Let $\alpha$ be an automorphism of $S$ of order
$m\neq 1$ in its action on $S/\gamma_{1}(S)$. Assume that there exists
$c\in\mathrm{GF}(p)$ such that $s_{j}\alpha\equiv
s_{j}^{c}\mod{\gamma_{j+1}(S)}$ and $s_{k}\alpha\equiv
s_{k}^{c}\mod{\gamma_{k+1}(S)}$ for some $j,k\geq 1$. Then $j\equiv
k\pmod{m}$.

###### Proof.

As $m$ and $p$ are coprime, there is $x\in S\setminus\gamma_{1}(S)$ such that
$\alpha$ leaves invariant $\langle x\rangle\gamma_{2}(S)$. Let
$a,b\in\mathrm{GF}(p)$ be such that $x\alpha\equiv x^{a}\mod\gamma_{2}(S)$ and
$s_{1}\alpha\equiv s_{1}^{b}\mod\gamma_{2}(S)$. By assumption there is
$c\in\mathrm{GF}(p)$ and $j,k\geq 1$ such that

$s_{j}\alpha\equiv s_{j}^{c}\mod\gamma_{j+1}(S)\text{ and }s_{k}\alpha\equiv
s_{k}^{c}\mod\gamma_{k+1}(S).$

Since $S$ is not exceptional, by Lemma 3.9 we have

$s_{j}\alpha\equiv s_{j}^{a^{j-1}b}\mod\gamma_{j+1}(S)\text{ and
}s_{k}\alpha\equiv s_{k}^{a^{k-1}b}\mod\gamma_{k+1}(S).$

Therefore

$a^{j-1}b\equiv c\equiv a^{k-1}b\pmod{p}.$

Thus $a^{j-k}\equiv 1\pmod{p}$, that is $j-k\equiv 0\pmod{m}$ and so

$j\equiv k\pmod{m}.$

∎

###  3.2. A theorem of Juhász

For completeness, we now present a proof of [35, Theorem 6.2] slightly
modified for our application.

######  Theorem 3.14 (Juhász).

Assume that $S$ has maximal class and order at least $p^{4}$. If
$\gamma_{3}(S)$ is abelian, then either $\gamma_{2}(S)$ has nilpotency class
at most $2$ or $|S|\geq p^{2p+4}$. In particular, if $\gamma_{j}(S)>1$ is
elementary abelian for some $j\geq 3$, then $\gamma_{j-1}(S)$ has nilpotency
class at most $2$.

###### Proof.

Let $k,\ell\geq 1$ be such that

$\gamma_{k}(S)=[\gamma_{3}(S),\gamma_{2}(S)]=\gamma_{2}(S)^{\prime}\text{ and
}\gamma_{\ell}(S)=[\gamma_{3}(S),\gamma_{2}(S),\gamma_{2}(S)].$

Assume that $\gamma_{2}(S)$ does not have nilpotency class $2$. Then
$\gamma_{\ell}(S)\neq 1$. We shall show that $|S|\geq p^{2p+4}$.

Observe that for $j\geq 3$,

$[\gamma_{j}(S),\gamma_{2}(S)]=\gamma_{j+k-3}(S).$

To see this, we note the case $j=3$ is just the definition of $k$. Assume that
the statement holds for $j\geq 3$. Then
$[S,\gamma_{2}(S),\gamma_{j}(S)]=[\gamma_{3}(S),\gamma_{j}(S)]=1$ because
$\gamma_{j}(S)\leq\gamma_{3}(S)$ and $\gamma_{3}(S)$ is abelian. Hence the
Three Subgroup Lemma yields the second equality of

$[\gamma_{j+1}(S),\gamma_{2}(S)]=[\gamma_{j}(S),S,\gamma_{2}(S)]=[\gamma_{2}(S),\gamma_{j}(S),S]=[\gamma_{j+k-3}(S),S]=\gamma_{(j+1)+k-3}(S)$

and this verifies the observation. In particular,
$\gamma_{\ell}(S)=[\gamma_{k}(S),\gamma_{2}(S)]=\gamma_{2k-3}(S)$ and so

$\ell=2k-3$ and $|S|=p^{n}\geq p^{2k-2}$.

Recall the definition of $s$ and $s_{i}$, $1\leq i\leq n-1$ from Subsection
3.1 and for $i\geq n$ we set $s_{i}=1$.

Define $u_{3}=[s_{2},s_{1}]\in\gamma_{3}(S)$ and
$u_{i}=[s,u_{i-1}]\in\gamma_{i}(S)$ for $i\geq 4$. Since
$\gamma_{2}(S)/\gamma_{k}(S)$ is abelian and $[s,s_{1}]\in\gamma_{2}(S)$,
using the variant of the Hall-Witt identity

$[x,y,z^{x}][y,z,x^{y}][z,x,y^{z}]=1$

and using $[s_{2}^{-1},s_{j}]\in Z(S/\gamma_{j+k-2}(S))$ we calculate

$\displaystyle\gamma_{j+k-2}(S)$
$\displaystyle=[s,s_{j},s_{1}^{s}][s_{j},s_{1},s^{s_{j}}][s_{1},s,s_{j}^{s_{1}}]\gamma_{j+k-2}(S)=[s_{j+1},s_{1}^{s}][s_{j},s_{1},s^{s_{j}}][s_{2}^{-1},s_{j}^{s_{1}}]\gamma_{j+k-2}(S)$
$\displaystyle=[s_{j+1},s_{1}[s_{1},s]][[s_{j},s_{1}]^{s_{j}^{-1}},s]^{s_{j}}[s_{2}^{-1},s_{j}]\gamma_{j+k-2}(S)=[s_{j+1},s_{1}s_{j}^{-1}][s_{j},s_{1},s][s_{j},s_{2}]\gamma_{j+k-2}(S)$
$\displaystyle=[s_{j+1},s_{1}][s_{j},s_{1},s][s_{j},s_{2}]\gamma_{j+k-2}(S).$

Hence

(1) $\displaystyle\;[s_{j+1},s_{1}][s_{j},s_{2}]\gamma_{j+k-2}(S)$
$\displaystyle=[s,[s_{j},s_{1}]]\gamma_{j+k-2}(S);\text{ and }$
$\displaystyle[s_{3},s_{1}]\gamma_{k}(S)$ $\displaystyle=u_{4}\gamma_{k}(S).$

Using $\gamma_{2}(S)/\gamma_{\ell}(S)$ has class $2$ and $\gamma_{3}(S)$ is
abelian together with the Hall-Witt identity and we have, for $t\geq 3$,

$\displaystyle\gamma_{\ell}(S)$ $\displaystyle=$
$\displaystyle[s,s_{t},s_{2}]^{s_{t}^{-1}}[s_{t}^{-1},s_{2}^{-1},s]^{s_{2}}[s_{2},s^{-1},s_{t}^{-1}]^{s}\gamma_{\ell}(S)=[s,s_{t},s_{2}][s_{t}^{-1},s_{2}^{-1},s]\gamma_{\ell}(S)$
$\displaystyle=$
$\displaystyle[s,s_{t},s_{2}][s_{t},s_{2},s]\gamma_{\ell}(S).$

So we know for $t\geq 3$

(2)
$[s_{t+1},s_{2}]\gamma_{\ell}(S)=[s_{t},s_{2},s]^{-1}\gamma_{\ell}(S)=[s,[s_{t},s_{2}]]\gamma_{\ell}(S).$

We claim that, for $j\geq 3$,

(3)
$[s_{j},s_{1}][s_{j-1},s_{2}]^{j-3}\gamma_{j+k-3}(S)\gamma_{\ell}(S)=u_{j+1}\gamma_{j+k-3}(S)\gamma_{\ell}(S).$

This is valid for $j=3$ by (1). We prove the claim by induction. We have

$[s,[s_{j},s_{1}][s_{j-1},s_{2}]^{j-3}]\gamma_{j+k-2}(S)\gamma_{\ell}(S)=[s,u_{j+1}]\gamma_{j+k-2}(S)\gamma_{\ell}(S)=u_{j+2}\gamma_{j+k-2}(S)\gamma_{\ell}(S).$
$\displaystyle[s,[s_{j},s_{1}][s_{j-1},s_{2}]^{j-3}]\gamma_{j+k-2}(S)\gamma_{\ell}(S)$
$\displaystyle=$
$\displaystyle[s,[[s_{j-1},s_{2}]^{j-3}][s,[s_{j},s_{1}]]^{[s_{j},s_{1}]^{j-3}}\gamma_{j+k-2}(S)\gamma_{\ell}(S)$
$\displaystyle=^{\gamma_{3}\text{ abelian }}$
$\displaystyle[s,[s_{j-1},s_{2}]]^{j-3}[s,[s_{j},s_{1}]]\gamma_{j+k-2}(S)\gamma_{\ell}(S)$
$\displaystyle=^{\text{(\ref{eq2}) and (\ref{eqnew})}}$
$\displaystyle[s_{j},s_{2}]^{j-2}[s_{j+1},s_{1}]\gamma_{j+k-2}(S)\gamma_{\ell}(S).$

Thus (3) follows by induction.

Write $[s_{3},s_{2}]=\prod_{j=k}^{n-1}s_{j}^{a_{j}}$ where $0\leq a_{j}\leq
p-1$ and $a_{k}\neq 0$. As $\gamma_{3}(S)$ is abelian, we obtain

$[s_{3},s_{2},s_{1}]=[\prod_{j=k}^{n-1}s_{j}^{a_{j}},s_{1}]=\prod_{j=k}^{n-1}[s_{j},s_{1}]^{a_{j}}.$

Hence, using equation (3), taking suitable $g_{j+k-3}\in\gamma_{j+k-3}(S)$
noting that $[s_{2},s_{j}]\in\gamma_{j+k-3}(S)\leq\gamma_{\ell}(S)$ for $j\geq
k$ and remembering $\gamma_{3}(S)$ is abelian, we calculate

(4)
$\displaystyle[s_{3},s_{2},s_{1}]\gamma_{\ell}(S)=\prod_{j=k}^{n-1}[s_{j},s_{1}]^{a_{j}}\gamma_{\ell}(S)=\prod_{j=k}^{n-1}(u_{j+1}[s_{2},s_{j-1}]^{j-3}g_{j+k-3})^{a_{j}}\gamma_{\ell}(S)$
$\displaystyle=[s_{2},s_{k-1}]^{a_{k}(k-3)}\prod_{j=k}^{n-1}u_{j+1}^{a_{j}}g_{j+k-3}^{a_{j}}\gamma_{\ell}(S)=[s_{2},s_{k-1}]^{a_{k}(k-3)}\prod_{j=k}^{n-1}u_{j+1}^{a_{j}}\gamma_{\ell}(S)$

where $g_{j+k-3}\in\gamma_{j+k-3}(S)\leq\gamma_{\ell}(S)$ as, for $j\geq k$,
$j+k-3\geq 2k-3\geq\ell$. We next determine
$\prod_{j=k}^{n-1}u_{j+1}^{a_{j}}\gamma_{\ell}(S).$ We decompose

$u_{3}=[s_{2},s_{1}]=\prod_{t=3}^{n}s_{t}^{b_{t}}.$

Then, since $\gamma_{3}(S)$ is abelian,

$u_{4}=[s,u_{3}]=[s,\prod_{t=3}^{n}s_{t}^{b^{t}}]=\prod_{t=3}^{n}[s,s_{t}]^{b_{t}}=\prod_{t=3}^{n}s_{t+1}^{b_{t}}.$

and by induction, for $r\geq 3$, $u_{r}=\prod_{t=3}^{n}s_{t+r-3}^{b_{t}}$.

Using induction and (2), for $t\geq 3$,
$[s,[s_{t},s_{2}]]\gamma_{\ell}(S)=\prod_{j=k}^{n}s_{j+t-2}^{a_{j}}\gamma_{\ell}(S)$
and so

$\displaystyle[u_{4},s_{2}]\gamma_{\ell}(S)$ $\displaystyle=$
$\displaystyle\prod_{t=3}^{n}[s_{t+1},s_{2}]^{b_{t}}\gamma_{\ell}(S)=\prod_{t=3}^{n}([s,[s_{t},s_{2}])^{b_{t}}\gamma_{\ell}(S)=\prod_{t=3}^{n}\left(\prod_{j=k}^{n}s_{j+t-2}^{a_{j}}\right)^{b_{t}}\gamma_{\ell}(S)$
$\displaystyle=$
$\displaystyle\prod_{j=k}^{n}\left(\prod_{t=3}^{n}s_{j+t-2}^{b_{t}}\right)^{a_{j}}\gamma_{\ell}(S)=\prod_{j=k}^{n}u_{j+1}^{a_{j}}\gamma_{\ell}(S).$

In combination with equation (4) this provides

(5)
$[s_{3},s_{2},s_{1}]\gamma_{\ell}(S)=[s_{2},s_{k-1}]^{a_{k}(k-3)}[u_{4},s_{2}]\gamma_{\ell}(S).$

On the other hand, commutating Equation (1) on the right with $s_{2}$ yields

(6)
$[s_{3},s_{1},s_{2}]\gamma_{\ell}(S)=[u_{4}g_{k},s_{2}]\gamma_{\ell}(S)=[u_{4},s_{2}]\gamma_{\ell}(S).$

Since
$[s_{3},s_{2},s_{1}]\gamma_{\ell}(S)=[s_{3},s_{1},s_{2}]\gamma_{\ell}(S)$, we
deduce that $[s_{2},s_{k-1}]^{a_{k}(k-3)}\in\gamma_{\ell}(S)$. However,

$\langle[s_{k-1},s_{2}]\rangle\gamma_{\ell}(S)=[\gamma_{k-1}(S),\gamma_{2}(S)]=\gamma_{2k-4}(S)>\gamma_{\ell}(S)=\gamma_{2k-3}(S),$

and therefore $[s_{k-1},s_{2}]\not\in\gamma_{\ell}(S)$ and so
$a_{k}(k-3)\equiv 0\pmod{p}$. Since $0<a_{k}\leq p-1$, $k-3\equiv 0\pmod{p}$
and, as $k>3$, $k-3=mp$ for some $m\geq 1$. Hence $n\geq 2k-2\geq 2p+4$. We
conclude, $|S|\geq p^{2p+4}$. This proves the main statement.

Assume that $\gamma_{j}(S)$ is non-trivial and elementary abelian for some
$j\geq 3$. Let $x\in S\setminus(\gamma_{1}(S)\cup C_{S}(Z_{2}(S)))$. Then
$x^{p}\in C_{S}(x)\cap\gamma_{1}(S)=Z(S)\leq\gamma_{2}(S)$ by Lemma 3.3. Hence
$T=\langle x\rangle\gamma_{j-2}(S)$ has order $p^{n-j+2}\geq p^{4}$. By Lemma
3.5, $T$ has maximal class and $\gamma_{j}(S)=\gamma_{3}(T)$. Hence
$\gamma_{3}(T)$ is elementary abelian. Lemma 3.1 (iv) implies
$|\gamma_{3}(T)|\leq p^{p-1}$ and thus $|T|\leq p^{p+2}$. Since $p+2<2p+4$,
the main claim yields $\gamma_{2}(T)=\gamma_{j-1}(S)$ has nilpotency class at
most $2$. This completes the proof. ∎

##  4\. Representations of groups with a cyclic Sylow $p$-subgroup

In this section, for $p$ an odd prime, we gather together various facts about
representations of groups with cyclic Sylow $p$-subgroups.

######  Definition 4.1.

A group $X$ is said to be of _$\mathrm{L}_{2}(p)$ -type_ provided each
composition factor of $X$ is either a $p$-group, a $p^{\prime}$-group or is
isomorphic to $\mathrm{PSL}_{2}(p)$.

We will require the following result due to Feit.

######  Theorem 4.2 (Feit).

Suppose that $p$ is a prime, $\mathbb{K}$ is a field of characteristic $p$,
$L$ is a finite group with a cyclic Sylow $p$-subgroup $P$ and $V$ is a
faithful indecomposable $\mathbb{K}L$-module with $d=\dim V\leq p$. Assume
that $L$ is not of $\mathrm{L}_{2}(p)$-type. Then $p$ is odd, $|P|=p$,
$V|_{P}$ is indecomposable, $C_{L}(P)=P\times Z(L)$ and
$d\geq\frac{2}{3}(p-1)$.

###### Proof.

This is [24, Theorem 1]. ∎

Theorem 4.2 illuminates the importance of representations of
$\mathrm{SL}_{2}(p)$. Some of the results in this section hold for an
arbitrary field $\mathbb{K}$ but our applications will only be for
$\mathbb{K}=\mathrm{GF}(p)$. We follow the standard construction of certain
irreducible modules for $\mathrm{GL}_{2}(\mathbb{K})$. Let $\mathbb{K}[x,y]$
be the polynomial ring in two commuting variables $x$ and $y$ and coefficients
in $\mathbb{K}$. Then, for $\left(\begin{smallmatrix}\alpha&\beta\\\
\gamma&\delta\end{smallmatrix}\right)\in\mathrm{GL}_{2}(\mathbb{K})$ and
$a,b\geq 0$ natural numbers, the extension linearly of

$x^{a}y^{b}\cdot\left(\begin{smallmatrix}\alpha&\beta\\\
\gamma&\delta\end{smallmatrix}\right)=(\alpha x+\beta y)^{a}(\gamma x+\delta
y)^{b}$

makes $\mathbb{K}[x,y]$ into a $\mathbb{K}\mathrm{GL}_{2}(\mathbb{K})$-module.
The subspaces of $\mathbb{K}[x,y]$ consisting of homogenous polynomials of
fixed degree at most $p-1$ provide us with an explicit construction of the
_basic_ irreducible $\mathbb{K}\mathrm{SL}_{2}(p)$-modules by restriction.

######  Notation 4.3.

For $0\leq e\leq p-1$, $\mathbf{V}_{e}$ represents the $(e+1)$-dimensional
$\mathrm{GL}_{2}(\mathbb{K})$-submodule of $\mathbb{K}[x,y]$ consisting of
degree $e$ homogeneous polynomials. We use the same notation for
$\mathbf{V}_{e}$ when we consider $\mathbf{V}_{e}$ as a module for certain
subgroups of $\mathrm{GL}_{2}(\mathbb{K})$, for example, when considered as a
$\mathbb{K}\mathrm{SL}_{2}(p)$-module. We often call $\mathbf{V}_{1}$ the
natural $\mathrm{SL}_{2}(p)$-module.

For $\mathrm{SL}_{2}(p)$, every irreducible
$\mathbb{K}\mathrm{SL}_{2}(p)$-module is basic and can be realized over
$\mathrm{GF}(p)$ (see [1, page 15] for example). In particular, in this case
there are $p$ irreducible modules and they have dimensions $1,2,\dots,p$. The
faithful modules are the ones of even-dimension and the odd-dimensional
modules are representations of $\mathrm{PSL}_{2}(p)$.

The next eight results provide the facts that we shall require about these
representations. The first result is used silently in the text.

######  Lemma 4.4.

Suppose $L\cong\mathrm{SL}_{2}(p)$, $T\in\hbox{\rm Syl}_{p}(L)$ and $V$ is an
irreducible $\mathrm{GF}(p)\mathrm{SL}_{2}(p)$-module. Then $V$ is
indecomposable as a $\mathrm{GF}(p)T$-module. In particular, for all $0\leq
k\leq\dim V-1$, $\dim[V,T;k]/[V,T;k+1]=1$ and, if $\dim[V,T]=1$, then
$V\cong\mathbf{V}_{1}$ has dimension $2$.

###### Proof.

This is calculated using the description of the modules above.∎

Obviously, if $p$ is odd and $V$ is a faithful $\mathrm{SL}_{2}(p)$-module,
then the centre of $\mathrm{SL}_{2}(p)$ negates $V$ and so a complement to a
Sylow $p$-subgroup of $\mathrm{SL}_{2}(p)$ acts fixed-point-freely on any
faithful module. The same is not true for the irreducible
$\mathrm{PSL}_{2}(p)$-modules.

######  Lemma 4.5.

Suppose that $L\cong\mathrm{SL}_{2}(p)$, $T\in\hbox{\rm Syl}_{p}(L)$ and $H$
is a complement to $T$ in $N_{L}(T)$. Assume that $V$ is an irreducible
$d$-dimensional $\mathrm{GF}(p)L$-module. If $C_{V}(H)\neq 0$, then $d$ is odd
and either

  1. (i)

$d\leq p-2$, $\dim C_{V}(H)=1$ and $[V,T;(d-1)/2]/[V,T;(d+1)/2]$ is
centralized by $H$; or

  2. (ii)

$d=p$, $\dim C_{V}(H)=3$ and $V/[V,T]$, $[V,T;(p-1)/2]/[V,T;(p+1)/2]$ and
$C_{V}(T)$ are centralized by $H$.

In particular, if $C_{V}(T)$ is centralized by $H$, then either $V$ is the
trivial module or $\dim V=p$ and $\dim C_{V}(H)=3$.

###### Proof.

Let $e=d-1$ and remember that $V=\mathbf{V}_{e}$. Take
$\tau=\left(\begin{smallmatrix}1&1\\\ 0&1\end{smallmatrix}\right)$ to be a
generator of $T$ and $\delta=\left(\begin{smallmatrix}\lambda&0\\\
0&\lambda^{-1}\end{smallmatrix}\right)$ where $\lambda\in\mathrm{GF}(p)$ has
order $p-1$ to be a generator of $H$. Then we calculate

$[V,T;k]=\langle x^{j}y^{e-j}\mid 0\leq j\leq e-k\rangle.$

We also calculate that $\delta$ acts as the scalar
$\lambda^{e-k}\lambda^{-k}=\lambda^{e-2k}$ on the quotient

$[V,T;k]/[V,T;k+1]=\langle x^{e-k}y^{k}+[V,T;k+1]\rangle.$

Hence $\tau$ centralizes $[V,T;k]/[V,T;k+1]$ if and only if either $k=e/2$ or
$e=p-1$ and $k=0$ or $p-1$. In particular, $e=d-1$ is even and this gives the
result. ∎

Assume that $\mathbb{K}$ has characteristic $p\geq 0$ and that
$\mathbf{V}_{d}$ is the $(d+1)$-dimensional
$\mathrm{GL}_{2}(\mathbb{K})$-module of homogeneous polynomials of degree $d$.
Define

$\displaystyle\Omega:\mathbb{K}[x,y]\otimes\mathbb{K}[x,y]$
$\displaystyle\rightarrow$
$\displaystyle\mathbb{K}[x,y]\otimes\mathbb{K}[x,y]$ $\displaystyle(f\otimes
g)$ $\displaystyle\mapsto$ $\displaystyle\frac{\partial f}{\partial
x}\otimes\frac{\partial g}{\partial y}-\frac{\partial f}{\partial
y}\otimes\frac{\partial g}{\partial x}.$

Then $\Omega$ is $\mathbb{K}$-linear. Let $A=\left(\begin{smallmatrix}1&1\\\
0&1\end{smallmatrix}\right)$ and
$B_{\theta}=\left(\begin{smallmatrix}0&-\theta\\\ 1&0\end{smallmatrix}\right)$
with $\theta\in\mathbb{K}^{\times}$ be elements of
$\mathrm{GL}_{2}(\mathbb{K})$. Then $\mathrm{GL}_{2}(\mathbb{K})=\langle
A,B_{\theta}\mid\theta\in\mathbb{K}^{\times}\rangle$. We calculate

$\displaystyle(x^{a}y^{b}\otimes x^{c}y^{d})A\Omega$ $\displaystyle=$
$\displaystyle((x+y)^{a}y^{b}\otimes(x+y)^{c}y^{d})\Omega$ $\displaystyle=$
$\displaystyle
a(x+y)^{a-1}y^{b}\otimes\left(c(x+y)^{c-1}y^{d}+d(x+y)^{c}y^{d-1}\right)$
$\displaystyle-\left(a(x+y)^{a-1}y^{b}+b(x+y)^{a}y^{b-1}\right)\otimes
c(x+y)^{c-1}y^{d}$ $\displaystyle=$ $\displaystyle
ad(x+y)^{a-1}y^{b}\otimes(x+y)^{c}y^{d-1}-bc(x+y)^{a}y^{b-1}\otimes(x+y)^{c-1}y^{d}$
$\displaystyle=$ $\displaystyle\left(adx^{a-1}y^{b}\otimes
x^{c}y^{d-1}-bcx^{a}y^{b-1}\otimes x^{c-1}y^{d}\right)A$ $\displaystyle=$
$\displaystyle(x^{a}y^{b}\otimes x^{c}y^{d})\Omega A.$

Hence $\Omega A=A\Omega.$ Similarly, we calculate that
$B_{\theta}\Omega=\theta\Omega B_{\theta}$ and so $C\Omega=(\det C)\Omega C$
for all $C\in\mathrm{GL}_{2}(\mathbb{K})$. In particular, $\Omega$ is a
$\mathbb{K}\mathrm{SL}_{2}(\mathbb{K})$-module homomorphism. The
multiplication map
$\mu:\mathbb{K}[x,y]\otimes\mathbb{K}[x,y]\rightarrow\mathbb{K}[x,y]$ defined
by $(f\otimes g)\mu\mapsto fg$ is also a
$\mathbb{K}\mathrm{SL}_{2}(\mathbb{K})$-module homomorphism. It follows that,
for $r\geq 0$, the _$r$ -transvectant_

$\Theta_{r}=\Omega^{r}\mu:\mathbb{K}[x,y]\otimes\mathbb{K}[x,y]\rightarrow\mathbb{K}[x,y]$

is also a $\mathbb{K}\mathrm{SL}_{2}(\mathbb{K})$-module homomorphism.

Suppose that $d$ and $e$ are natural numbers and observe that by restriction

$\Theta_{r}:\mathbf{V}_{d}\otimes\mathbf{V}_{e}\rightarrow\mathbf{V}_{d+e-2r}.$

Assume now that $\mathbb{K}$ has characteristic $p>0$. Then for $\ell\leq
p-1$, $\mathbf{V}_{\ell}$ is an irreducible
$\mathrm{SL}_{2}(\mathbb{K})$-module. Therefore, if $d+e-2r\leq p-1$, the
restriction of $\Theta_{r}$ is either the zero map or is a surjection. In
particular, if $r\leq e\leq d$ and $d+e\leq p-1$, then, as $(x^{d}\otimes
y^{e})\Theta_{r}\neq 0$, $\Theta_{r}$ restricts to a surjection. By counting
dimensions, we conclude

$\mathbf{V}_{d}\otimes\mathbf{V}_{e}\cong\mathbf{V}_{d+e}\oplus\mathbf{V}_{d+e-2}\oplus\dots\oplus\mathbf{V}_{d-e}$

which in characteristic $0$ is known as the _Clebsch-Gordan decomposition_.
Let $\iota$ be the $\mathbb{K}\mathrm{SL}_{2}(\mathbb{K})$-module endomorphism
$\mathbb{K}[x,y]$ which maps $f\otimes g$ to $g\otimes f$. Then
$\Theta_{r}\iota=(-1)^{r}\Theta_{r}$ and, for a fixed natural number $d$,
$\Lambda^{2}(\mathbf{V}_{d})$ is the submodule of
$\mathbf{V}_{d}\otimes\mathbf{V}_{d}$ negated by $\iota$ and
$S^{2}(\mathbf{V}_{d})$ is the submodule centralized by $\iota$. We have
recreated the following

######  Proposition 4.6.

Suppose that $d,e\in\mathbb{N}$ with $d\geq e$ and $d+e\leq p-1$. Then, as
$\mathrm{SL}_{2}(\mathbb{K})$-modules,

$\mathbf{V}_{d}\otimes\mathbf{V}_{e}\cong\mathbf{V}_{d+e}\oplus\mathbf{V}_{d+e-2}\oplus\dots\oplus\mathbf{V}_{d-e}.$

Furthermore, if $2d\leq p-1$,

$S^{2}(\mathbf{V}_{d})\cong\mathbf{V}_{2d}\oplus\mathbf{V}_{2d-4}\oplus\dots\oplus\mathbf{V}_{a}$

and

$\Lambda^{2}(\mathbf{V}_{d})=\mathbf{V}_{2d-2}\oplus\mathbf{V}_{2d-6}\oplus\dots\oplus\mathbf{V}_{b}$

where $a=2d\pmod{4}$ and $b=2d-2\pmod{4}$ with $a,b\leq 2$.∎

Notice that, if we take $d+1=(p-1)/2$, then $\mathbf{V}_{d}$ is involved in
$\Lambda^{2}(\mathbf{V}_{d})$ if and only if $d\equiv 2\pmod{4}$.

######  Lemma 4.7.

We have

$\mathbf{V}_{p-3}\otimes\mathbf{V}_{p-3}=\mathbf{V}_{0}\oplus\mathbf{V}_{2}\oplus\mathbf{V}_{p-1}\oplus
P(p-3)\oplus\dots\oplus P(4)$

where, for $1\leq j\leq p-1$, $P(j)$ is the projective cover of
$\mathbf{V}_{j}$. In particular,
$(\mathbf{V}_{p-3}\otimes\mathbf{V}_{p-3})/\mathrm{Rad}(\mathbf{V}_{p-3}\otimes\mathbf{V}_{p-3})$
is the direct sum of all the irreducible $\mathrm{PSL}_{2}(p)$-modules.

###### Proof.

It suffices to work over an algebraically closed field. From [18, Lemma 3.1
(ii)], we have

$\mathbf{V}_{p-3}\otimes\mathbf{V}_{p-3}=\mathbf{V}_{0}\oplus\mathbf{V}_{2}\oplus\mathbf{V}_{p-1}\oplus
T(p+1)\oplus\dots\oplus T(2p-6)$

where, for an integer $j$, $T(j)$ is the _tilting module_ associated to $j$.
We know $\mathbf{V}_{p-1}$ has dimension $p$ and is a projective module (see
[1]). Using [18, Lemma 3.1 (iii)]

$\mathbf{V}_{p-3}\otimes\mathbf{V}_{p-1}=\mathbf{V}_{p-1}\oplus
T(p+1)\oplus\dots\oplus T(2p-4).$

Since $\mathbf{V}_{p-1}$ is projective, so is
$\mathbf{V}_{p-3}\otimes\mathbf{V}_{p-1}$ and every direct summand of
$\mathbf{V}_{p-3}\otimes\mathbf{V}_{p-1}$ by [7, Lemma 1.5.2]. Hence the
tilting modules $T(p+1),\dots,T(2p-6)$ are projective
$\mathrm{SL}_{2}(p)$-modules. Now, the discussion before [18, Lemma 3.1]
reveals that for $p+1\leq j\leq 2p-2$, $T(j)$ has a quotient
$\mathbf{V}_{2(p-1)-j}$ and dimension $2p$. We conclude that $T(j)$ is the
projective cover $P(2(p-1)-j)$ of $\mathbf{V}_{2(p-1)-j}$. With this notation,
we have

$\mathbf{V}_{p-3}\otimes\mathbf{V}_{p-3}=\mathbf{V}_{0}\oplus\mathbf{V}_{2}\oplus\mathbf{V}_{p-1}\oplus
P(p-3)\oplus\dots\oplus P(4).$

In particular, we note that every irreducible $\mathrm{PSL}_{2}(p)$-module
appears exactly once as a quotient of
$(\mathbf{V}_{p-3}\otimes\mathbf{V}_{p-3})/\mathrm{Rad}(\mathbf{V}_{p-3}\otimes\mathbf{V}_{p-3})$.
This proves the claim. ∎

We recall from [1, pages 15 and 48] that $P(0)$ has dimension $p$ and is a
uniserial $\mathrm{PSL}_{2}(p)$-module with composition factors
$\mathbf{V}_{0}$, $\mathbf{V}_{p-3}$ and $\mathbf{V}_{0}$. In particular,
there is a unique indecomposable $\mathrm{PSL}_{2}(p)$-module with socle of
dimension $1$ and quotient $\mathbf{V}_{p-3}$.

######  Lemma 4.8.

Suppose that $L\cong\mathrm{PSL}_{2}(p)$ and $W$ is the unique indecomposable
$\mathrm{GF}(p)L$-module with socle of dimension $1$ and quotient
$\mathbf{V}_{p-3}$. Then $\Lambda^{2}(W)$ has a submodule
$U\leq\mathrm{Rad}(\Lambda^{2}(W))$ with $U\cong\mathbf{V}_{p-3}$ and
$\Lambda^{2}(W)/U\cong\Lambda^{2}(\mathbf{V}_{p-3})$.

###### Proof.

Let $R$ be the socle of $W$. Then $W\otimes W$ has a submodule

$U^{*}=\langle r\otimes w,w\otimes r\mid r\in R,w\in W\rangle.$

Plainly $W/U^{*}\cong\mathbf{V}_{p-3}\otimes\mathbf{V}_{p-3}$. Set $U=\langle
w\otimes r-r\otimes w\mid r\in R,w\in W\rangle$. Then $U\cong
W/R=\mathbf{V}_{p-3}$ and
$\Lambda^{2}(W)/U\cong\Lambda^{2}(\mathbf{V}_{p-3})$. Hence we only need to
show that $U\leq\mathrm{Rad}(\Lambda^{2}(W))$. Since $\Lambda^{2}(W)$ is a
direct summand of $W\otimes W$, we have
$\mathrm{Rad}(\Lambda^{2}(W))=\mathrm{Rad}(W\otimes W)\cap\Lambda^{2}(W)$. In
particular, if $U\not\leq\mathrm{Rad}(\Lambda^{2}(W))$, then $U$ is a direct
summand of $W\otimes W$. So suppose that this is the case. Then $W\otimes
W\cong\mathbf{V}_{p-3}\oplus(\mathbf{V}_{p-3}\otimes\mathbf{V}_{p-3})$. Let
$T\in\mathrm{Syl}_{p}(L)$. Then using Lemma 4.7, as a $T$-module
$\mathbf{V}_{p-3}\oplus(\mathbf{V}_{p-3}\otimes\mathbf{V}_{p-3})$ is a sum of
indecomposable modules $\mathbf{V}_{0}$ of dimension 1, $\mathbf{V}_{2}$ of
dimension $3$, $\mathbf{V}_{p-3}$ of dimension $p-2$ and a number of free
modules of dimension $p$. On the other hand, as $W$ is indecomposable with
socle $\mathbf{V}_{0}$ and quotient $\mathbf{V}_{p-3}$, using Lemma 4.4 and
[7, Corollary 3.6.10]we have that $W$ restricted to $T$ is indecomposable of
dimension $p-1$. It follows that, as a $T$-module, $W\otimes W$ is a direct
sum of a trivial module and a free module (for example use [18, Lemma 3.1
(ii)] to write down $\mathbf{V}_{p-2}\otimes\mathbf{V}_{p-2}$ and then
restrict to $T$). Since the two structures are incompatible, we conclude that
$U\leq\mathrm{Rad}(\Lambda^{2}(W))$. ∎

We can now establish the technical point that we require.

######  Lemma 4.9.

Suppose that $L\cong\mathrm{PSL}_{2}(p)$, $T\in\hbox{\rm Syl}_{p}(L)$, $H\leq
N_{L}(T)$ is a complement to $T$ and $W$ is the unique indecomposable
$\mathrm{GF}(p)L$-module with socle of dimension $1$ and quotient
$\mathbf{V}_{p-3}$. Assume that $\theta:W\times W\rightarrow\mathbf{V}_{p-3}$
is a surjective alternating $L$-invariant bilinear map. Let $u\in
W\setminus[W,T]$ and $w\in[W,T;p-3]\setminus C_{W}(T)$ be such that $\langle
w\rangle$ and $\langle u\rangle$ are $H$-invariant. Then $(u,w)\theta\neq 0$.

###### Proof.

Since $\theta:W\times W\rightarrow\mathbf{V}_{p-3}$ is a surjective
$L$-invariant bilinear map, there is a unique surjective
$\mathrm{GF}(p)L$-module homomorphism $\widetilde{\theta}$ from
$\Lambda^{2}(W)$ to $\mathbf{V}_{p-3}$. Using Lemma 4.8, we have that
$\widetilde{\theta}$ determines a surjective homomorphisms $\theta^{*}$ from
$\Lambda^{2}(\mathbf{V}_{p-3})$ to $\mathbf{V}_{p-3}$. By Lemma 4.7,
$\Lambda^{2}(\mathbf{V}_{p-3})$ either has no quotient isomorphic to
$\mathbf{V}_{p-3}$, which is against our assumption that $\theta$ is
surjective, or $\ker\theta^{*}$ is the unique maximal submodule of
$\Lambda^{2}(\mathbf{V}_{p-3})$ which has quotient $\mathbf{V}_{p-3}$. Now the
$(p-3)/2$-transvectant $\Theta_{(p-3)/2}$ restricted to
$\Lambda^{2}(\mathbf{V}_{p-3})$ also has image in $\mathbf{V}_{p-3}$. Hence
$\ker\theta^{*}=\ker\Theta_{(p-3)/2}$.

We may take $u=x^{p-3}$ and $w=y^{p-3}$ in $\mathbf{V}_{p-3}$. We calculate
$(u\otimes
w)\Theta_{(p-3)/2}=\left({(p-3)}\atop{(p-3)/2}\right)^{2}(xy)^{(p-3)/2}\neq
0$. Therefore $(u,w)\not\in\ker\Theta_{(p-3)/2}=\ker\theta^{*}$. This shows
that $(u,w)\theta\neq 0$. ∎

In the next lemma we are interested in modules for $L=\mathrm{Sym}(p)$ defined
in characteristic $p$. The notation $S^{\lambda}$ denotes the _Specht module_
for $L$ corresponding to the partition $\lambda$ of $p$. The module
$D^{\lambda}$ is the unique irreducible quotient of $S^{\lambda}$ (see [34]).
Thus $S^{p-1,1}$ is a characteristic $p$ representation of $L$ of dimension
$p-1$ and can be identified with the submodule of the natural
$\mathrm{GF}(p)L$-permutation module $\langle v_{i}\mid 1\leq i\leq p\rangle$
which is the kernel of the augmentation map
$\sum_{i=1}^{p}\lambda_{i}v_{i}\mapsto\sum_{i=1}^{p}\lambda_{i}$. In this
case, $D^{p-1,1}=S^{p-1,1}/\langle\sum_{i=1}^{p}v_{i}\rangle$ has dimension
$p-2$. The result we shall need is as follows.

######  Lemma 4.10.

Suppose that $p$ is a prime, $L\cong\mathrm{Sym}(p)$ and $V=S^{p-1,1}$
considered as a $\mathrm{GF}(p)L$-module. Then $\Lambda^{2}(V)=S^{p-2,1^{2}}$
has irreducible composition factors $D^{p-1,1}$ and $D_{p-2,1^{2}}$ both with
multiplicity $1$ and, furthermore, $\Lambda^{2}(V)$ has no quotient of
dimension $1$ or isomorphic to $D^{p-1,1}$ .

###### Proof.

For prime $p\geq 11$, the isomorphism type and composition factors of
$\Lambda^{2}V$ are explicitly given in [38, Section 2]. The fact that
$D^{p-1,1}$ is not a quotient of $\Lambda^{2}(V)$ follows from [34, Corollary
12.2]. For $p=5$ and $p=7$, we have checked the assertion by computer. ∎

######  Lemma 4.11.

Assume that $X=\mathrm{Sym}(p)$, $Y=X^{\prime}=\mathrm{Alt}(p)$ and
$V=D^{p-1,1}$ is the $p-2$-dimensional module. Let $W=V|_{Y}$. Then
$\dim\mathrm{H}^{1}(Y,W)=1$.

###### Proof.

Let $\tau=(1,2,3)$ and $\sigma=(3,\dots,p)$. Then
$H=\langle\tau,\sigma\rangle$ acts transitively on $\Omega=\\{1,\dots,p\\}$
and, as $p$ is a prime, it is primitive. Since $H$ contains a $3$-cycle, $Y=H$
by Jordan’s Theorem [33, II.4.5]. If $U$ is a $\mathrm{GF}(p)Y$-module with
$W$ of codimension $2$ and $[U,Y]=W$, then $\dim C_{U}(\tau)=2+\dim
C_{W}(\tau)=2+p-4=p-2$ and $\dim C_{U}(\sigma)=2+\dim C_{W}(\sigma)=2+1=3$.
Hence $\dim C_{U}(H)\geq 1$ and this proves $\dim\mathrm{H}^{1}(Y,W)\leq 1$.
Since the natural $p$-point permutation module has a quotient $T$ with
$[T,Y]=W$ and $C_{T}(Y)=0$, we have $\dim\mathrm{H}^{1}(Y,W)=1$. ∎

##  5\. A primer on fusion systems

We assume some basic familiarity with fusion systems and recommend the
references [5, 17] as introductory texts. We follow the notation from these
sources. We start by flying over the standard definitions and at the same time
introduce some of the standard terminology from [5, 17].

######  Definition 5.1.

For a finite group $G$ and any subgroups $H,K\leq G$, define

$\mathrm{Hom}_{G}(H,K)=\\{\varphi\in\mathrm{Hom}(H,K)\mid\varphi=c_{g}\text{
for some }g\in G\text{ such that }H^{g}\leq K\\}$

and we set $\mathrm{Aut}_{G}(H)=\mathrm{Hom}_{G}(H,H)\cong N_{G}(H)/C_{G}(H)$.

More generally, if $H,K\leq G$, then we define
$\mathrm{Aut}_{K}(H)=N_{K}(H)/C_{K}(H)\cong(H\cap N_{G}(K))/(H\cap C_{G}(K))$
the group of automorphisms of $H$ induced by conjugation by elements of $K$
which normalise $K$. Visibly $\mathrm{Aut}_{K}(H)\leq\mathrm{Aut}(H)$.
Similarly, if $K\leq L$ and $A\leq\mathrm{Aut}(L)$, we write
$\mathrm{Aut}_{A}(K)$ to represent the group of automorphisms of $K$ generated
by the restriction of automorphisms in $N_{A}(K)=\\{\gamma\in A\mid
L\gamma=L\\}$. In this case
$\mathrm{Out}_{A}(K)=\mathrm{Aut}_{A}(K)\mathrm{Inn}(K)/\mathrm{Inn}(K)$.

######  Definition 5.2.

A _fusion system_ on a $p$-group $S$ is a category $\mathcal{F}$, with objects
the set of all subgroups of $S$, and morphisms injective group homomorphisms
which satisfy the following two properties for all $P,Q\leq S$:

  1. (i)

$\mathrm{Hom}_{S}(P,Q)\subseteq\mathrm{Mor}_{\mathcal{F}}(P,Q)\subseteq\mathrm{Inj}(P,Q)$;
and

  2. (ii)

each $\varphi\in\mathrm{Mor}_{\mathcal{F}}(P,Q)$ is the composite of an
$\mathcal{F}$-isomorphism followed by an inclusion.

######  Definition 5.3.

Suppose that $\mathcal{F}$ is a fusion system on a finite $p$-group $S$ and
$P\leq S$. Then

  1. (i)

the $\mathcal{F}$-_conjugacy class_ of $P$, is
$P^{\mathcal{F}}=\\{P\alpha\mid\alpha\in\mathrm{Hom}_{\mathcal{F}}(P,S)\\}$;

  2. (ii)

for $R\in P^{\mathcal{F}}$, $\alpha\in\mathrm{Hom}_{\mathcal{F}}(R,P)$,
$\alpha^{*}$ is the isomorphism between $\mathrm{Aut}_{\mathcal{F}}(R)$ and
$\mathrm{Aut}_{\mathcal{F}}(P)$ defined by
$\gamma\mapsto\alpha^{-1}\gamma\alpha$;

  3. (iii)

$P$ is _fully $\mathcal{F}$-normalized_ if and only if
$|N_{S}(P)|\geq|N_{S}(R)|$ for all $R\in P^{\mathcal{F}}$;

  4. (iv)

$P$ is _fully $\mathcal{F}$-centralized_ if and only if
$|C_{S}(P)|\geq|C_{S}(R)|$ for all $R\in P^{\mathcal{F}}$;

  5. (v)

$P$ is _centric_ if and only if $C_{S}(P)=Z(P)$, and $P$ is _$\mathcal{F}$
-centric_ if and only if $R$ is centric for all $R\in P^{\mathcal{F}}$;

  6. (vi)

if $R\in P^{\mathcal{F}}$ and $\alpha\in\mathrm{Hom}_{\mathcal{F}}(R,P)$,

$N_{\alpha}=\\{g\in
N_{S}(R)\mid\alpha^{-1}c_{g}\alpha\in\mathrm{Aut}_{S}(P)\\}$

is the $\alpha$-_extension control subgroup_ of $S$;

  7. (vii)

$P$ is $\mathcal{F}$-_receptive_ if and only if for all $R\in P^{\mathcal{F}}$
and $\alpha\in\mathrm{Hom}_{\mathcal{F}}(R,P)$, there exists
$\widetilde{\alpha}\in\mathrm{Hom}_{\mathcal{F}}(N_{\alpha},S)$ such that
$\widetilde{\alpha}|_{R}=\alpha;$

  8. (viii)

$P$ is _fully $\mathcal{F}$-automized_ if and only if
$\mathrm{Aut}_{S}(P)\in\hbox{\rm Syl}_{p}(\mathrm{Aut}_{\mathcal{F}}(P))$;

  9. (ix)

$P$ is $\mathcal{F}$-_saturated_ provided there exists $R\in P^{\mathcal{F}}$
such that $R$ is simultaneously

    1. (a)

fully $\mathcal{F}$-automized; and

    2. (b)

$\mathcal{F}$-receptive.

  10. (x)

$\mathcal{F}$ is _saturated_ if every subgroup of $S$ is
$\mathcal{F}$-saturated.

In our arguments, an important role is played by normalizer fusion systems.
Suppose that $\mathcal{F}$ is a saturated fusion system on $S$ and $T\leq S$.
Then $N_{\mathcal{F}}(T)$ is the fusion system on $N_{S}(T)$ with, for
$P,Q\leq N_{S}(T)$, $\mathrm{Hom}_{N_{\mathcal{F}}(T)}(P,Q)$ consists of
morphisms $\alpha\in\mathrm{Hom}_{\mathcal{F}}(P,Q)$ such that there exists
$\widetilde{\alpha}\in\mathrm{Hom}_{\mathcal{F}}(PT,QT)$ with
$T\widetilde{\alpha}=T$ and $\alpha=\widetilde{\alpha}|_{P}$. Importantly, if
$T$ is fully $\mathcal{F}$-normalized, then $N_{\mathcal{F}}(T)$ is saturated
[5, Theorem I.5.5].

The subgroup $O_{p}(\mathcal{F})$ is the product of all subgroups $T\leq S$
such that $N_{\mathcal{F}}(T)=\mathcal{F}$. A saturated fusion system
$\mathcal{F}$ is _constrained_ provided $O_{p}(\mathcal{F})$ is
$\mathcal{F}$-centric.

######  Theorem 5.4 (The Model Theorem).

Let $\mathcal{F}$ be a constrained fusion system. Then there exists a finite
group $G$ with $O_{p^{\prime}}(G)=1$, $S\in\hbox{\rm Syl}_{p}(G)$,
$\mathcal{F}=\mathcal{F}_{S}(G)$ and $C_{G}(O_{p}(G))\leq O_{p}(G)$.

###### Proof.

See [5, Theorem III.5.10]. ∎

We now gather some elementary consequences of the definitions above. We have
the following well-known fact and, as we shall use it several times, we
provide the proof.

######  Lemma 5.5.

Suppose that $\mathcal{F}$ is a fusion system on $S$ and $P\leq S$.

  1. (i)

If $\alpha\in N_{\mathrm{Aut}_{\mathcal{F}}(P)}(\mathrm{Aut}_{S}(P))$, then
$N_{\alpha}={N}_{S}(P)$.

  2. (ii)

If $P\leq S$ is $\mathcal{F}$-receptive, then
$N_{\mathrm{Aut}_{\mathcal{F}}(P)}(\mathrm{Aut}_{S}(P))=\\{\alpha|_{P}\mid\alpha\in
N_{\mathrm{Aut}_{\mathcal{F}}(N_{S}(P))}(P)\\}$.

###### Proof.

Part (i) is just the definition of $N_{\alpha}$.

Suppose that $\alpha\in
N_{\mathrm{Aut}_{\mathcal{F}}(P)}(\mathrm{Aut}_{S}(P))$. Then
$N_{\alpha}=N_{S}(P)$ by (i). Since $P$ is $\mathcal{F}$-receptive,
$\alpha=\widetilde{\alpha}|_{P}$ where
$\widetilde{\alpha}\in\mathrm{Hom}_{\mathcal{F}}(N_{S}(P),S)$. As
$P\widetilde{\alpha}=P\alpha=P$, we know
${N}_{S}(P)\widetilde{\alpha}=N_{S}(P\widetilde{\alpha})=N_{S}(P)$. Hence
$\widetilde{\alpha}\in N_{\mathrm{Aut}_{\mathcal{F}}(N_{S}(P))}(P)$ and so
$N_{\mathrm{Aut}_{\mathcal{F}}(P)}(\mathrm{Aut}_{S}(P))\subseteq\\{\alpha|_{P}\mid\alpha\in
N_{\mathrm{Aut}_{\mathcal{F}}(N_{S}(P))}(P)\\}$. Conversely, assume $g\in
N_{S}(P)$. Then $c_{g}\in\mathrm{Aut}_{S}(P)$ and, for $\beta\in
N_{\mathrm{Aut}_{\mathcal{F}}(N_{S}(P))}(P)$, we have
$(\beta|_{P})^{-1}c_{g}\beta|_{P}=c_{g\beta}\in\mathrm{Aut}_{S}(P)$. Hence
$\beta|_{P}\in N_{\mathrm{Aut}_{\mathcal{F}}(P)}(\mathrm{Aut}_{S}(P))$. This
proves (ii). ∎

######  Lemma 5.6.

Suppose that $\mathcal{F}$ is a fusion system on $S$, $T\leq S$ and
$\mathcal{K}=N_{\mathcal{K}}(T)$ is a subfusion system of $\mathcal{F}$ on
$N_{S}(T)$. Assume that $R\leq T$ is
$\mathrm{Aut}_{\mathcal{K}}(T)$-invariant. Then $\mathcal{K}\subseteq
N_{\mathcal{F}}(R)$. In particular, if $R\leq T$ is
$\mathrm{Aut}_{\mathcal{F}}(T)$-invariant, then $N_{\mathcal{F}}(T)\subseteq
N_{\mathcal{F}}(R)$.

###### Proof.

Suppose that $X,Y\leq N_{S}(T)$ and
$\theta\in\mathrm{Hom}_{\mathcal{K}}(X,Y)$. Then, as $R$ is
$\mathrm{Aut}_{\mathcal{K}}(T)$-invariant, it is also
$\mathrm{Aut}_{S}(T)$-invariant and thus $R$ is normalized by $N_{S}(T)$. In
particular, $X,Y\leq N_{S}(T)\leq N_{S}(R)$ and so $X$ and $Y$ are objects in
${N}_{\mathcal{F}}(R)$. Since $\theta\in\mathrm{Hom}_{\mathcal{K}}(X,Y)$ and
$\mathcal{K}=N_{\mathcal{K}}(T)$, the morphism $\theta$ extends to
$\hat{\theta}\in\mathrm{Hom}_{\mathcal{K}}(XT,YT)$ so that $T\hat{\theta}=T$.
Thus $\hat{\theta}|_{T}\in\mathrm{Aut}_{\mathcal{K}}(T)$ and $R\hat{\theta}=R$
because $R$ is $\mathrm{Aut}_{\mathcal{K}}(T)$-invariant. As $\hat{\theta}$
extends $\theta$, $\hat{\theta}|_{XR}\in\mathrm{Hom}_{\mathcal{F}}(XR,YR)$
also extends $\theta$ and this means that
$\theta\in\mathrm{Hom}_{N_{\mathcal{F}}(R)}(X,Y)$. Hence
$\mathrm{Hom}_{\mathcal{K}}(X,Y)\subseteq\mathrm{Hom}_{N_{\mathcal{F}}(R)}(X,Y)$
and this proves the main statement of the lemma.

Taking $\mathcal{K}=N_{\mathcal{F}}(T)$ and noting
$\mathrm{Aut}_{N_{\mathcal{F}}(T)}(T)=\mathrm{Aut}_{\mathcal{F}}(T)$, yields
the remaining statement. ∎

If $Q$ is normal in $\mathcal{F}$, then the _factor system_ $\mathcal{F}/Q$
has objects $\\{T/Q\mid Q\leq S\\}$ and, for $Q\leq T,R\leq S$, morphisms
$\mathrm{Hom}_{\mathcal{F}/Q}(T/Q,R/Q)=\\{\bar{\phi}\mid\phi\in\mathrm{Hom}_{\mathcal{F}}(T,R)\\}$
where $(tQ)\bar{\phi}=t\phi Q$ for $\phi\in\mathrm{Hom}_{\mathcal{F}}(T,R)$.

######  Lemma 5.7.

If $\mathcal{F}$ is saturated and $Q$ is normal in $\mathcal{F}$, then
$\mathcal{F}/Q$ is saturated

###### Proof.

This is [5, Lemma II.5.5]. ∎

######  Definition 5.8.

Suppose that $\mathcal{F}$ is a fusion system. A subgroup $P$ of $S$ is
_$\mathcal{F}$ -essential_ if $P<S$, $P$ is $\mathcal{F}$-centric, fully
$\mathcal{F}$-normalized and $\mathrm{Out}_{\mathcal{F}}(P)$ contains a
strongly $p$-embedded subgroup. We write $\mathcal{E}_{\mathcal{F}}$ to denote
the set of $\mathcal{F}$-essential subgroups of $\mathcal{F}$.

The main tool for classifying fusion systems is provided by the following
lemma.

######  Theorem 5.9 (Alperin-Goldschmidt).

If $\mathcal{F}$ is a saturated fusion system on the $p$-group $S$, then

$\mathcal{F}=\langle\mathrm{Aut}_{\mathcal{F}}(S),\mathrm{Aut}_{\mathcal{F}}(E)\mid
E\in\mathcal{E}_{\mathcal{F}}\rangle.$

###### Proof.

See [5, Theorem I.3.5]. ∎

######  Lemma 5.10.

Suppose that $\mathcal{F}$ is a saturated fusion system on $S$ and $E$ is an
$\mathcal{F}$-essential subgroup. Then $O_{p}(\mathcal{F})\leq E$ and
$O_{p}(\mathcal{F})$ is $\mathrm{Aut}_{\mathcal{F}}(E)$-invariant.

###### Proof.

Since $N_{O_{p}(\mathcal{F})}(E)$ is $\mathrm{Aut}_{\mathcal{F}}(E)$-invariant
and $O_{p}(\mathrm{Aut}_{\mathcal{F}}(E))=\mathrm{Inn}(E)$, we have
$N_{O_{p}(\mathcal{F})}(E)\leq E$ by Lemma 2.6. Hence $O_{p}(\mathcal{F})\leq
E$ and so is normal in $E$ and $\mathrm{Aut}_{\mathcal{F}}(E)$-invariant. ∎

We will also meet the fusion subsystems $O^{p}(\mathcal{F})$ and
$O^{p^{\prime}}(\mathcal{F})$. We define the _focal_ and _hyperfocal_
subgroups as follows

$\mathrm{foc}(\mathcal{F})=\langle[g,\alpha]\mid g\in Q\leq S\text{ and
}\alpha\in\mathrm{Aut}_{\mathcal{F}}(Q)\rangle$

$\mathrm{hyp}(\mathcal{F})=\langle[g,\alpha]\mid g\in Q\leq S\text{ and
}\alpha\in O^{p}(\mathrm{Aut}_{\mathcal{F}}(Q))\rangle$

where $[g,\alpha]=g^{-1}(g)\alpha$. The subfusion system $O^{p}(\mathcal{F})$
is a fusion system on $\mathrm{hyp}(\mathcal{F})$ defined as follows:

$O^{p}(\mathcal{F})=\langle\mathrm{Inn}(\mathrm{hyp}(\mathcal{F})),O^{p}(\mathrm{Aut}_{\mathcal{F}}(Q))\mid
Q\leq\mathrm{hyp}(\mathcal{F})\rangle.$

######  Lemma 5.11.

Let $\mathcal{F}$ be a saturated fusion system on $S$. The following hold:

  1. (i)

$\mathrm{foc}(\mathcal{F})=\langle[g,\alpha]\mid g\in Q\leq S,Q\text{ is
}\mathcal{F}\text{-essential or
}Q=S,\alpha\in\mathrm{Aut}_{\mathcal{F}}(Q)\rangle$.

  2. (ii)

$O^{p}(\mathcal{F})=\mathcal{F}$ if and only if $\mathrm{foc}(\mathcal{F})=S$
if and only if $\mathrm{hyp}(\mathcal{F})=S$.

###### Proof.

This follows from Theorem 5.9 and [5, Corollary I.7.5]. ∎

The subfusion system $O^{p^{\prime}}(\mathcal{F})$ is more complicated to
define, we just settle for saying that it is the unique saturated subfusion
system on $S$ minimal subject to containing
$O^{p^{\prime}}(\mathrm{Aut}_{\mathcal{F}}(Q))$ for all $Q\leq S$ (see [5,
Definition I.7.3]).

A saturated fusion system is call _reduced_ if and only if
$O_{p}(\mathcal{F})=1$ and
$\mathcal{F}=O^{p}(\mathcal{F})=O^{p^{\prime}}(\mathcal{F})$. Normal subfusion
systems are defined in [5, Definition I.6.1] and a _simple_ saturated fusion
system is a fusion system which contain no proper normal subfusion system.
Simple fusion systems are reduced as $O^{p}(\mathcal{F})$,
$O^{p^{\prime}}(\mathcal{F})$ and
$\mathcal{F}_{O_{p}(\mathcal{F})}(O_{p}(\mathcal{F}))$ are normal subsystems
of $\mathcal{F}$.

######  Lemma 5.12.

Suppose that $\mathcal{F}$ is a saturated fusion system on a $p$-group $S$.
Assume that each $P\in\mathcal{E}_{\mathcal{F}}$ is minimal among all
$\mathcal{F}$-centric subgroups. For each $P\in E_{\mathcal{F}}$ define

$\mathrm{Aut}^{(P)}_{\mathcal{F}}(S)=\langle\alpha\in\mathrm{Aut}_{\mathcal{F}}(S)\mid
P\alpha=P,\alpha|_{P}\in
O^{p^{\prime}}(\mathrm{Aut}_{\mathcal{F}}(P))\rangle.$

Then $O^{p^{\prime}}(\mathcal{F})=\mathcal{F}$ if and only if
$\mathrm{Aut}_{\mathcal{F}}(S)=\langle\mathrm{Inn}(S),\mathrm{Aut}_{\mathrm{G}}^{(P)}(S)\mid
P\in\mathcal{E}_{\mathcal{F}}\rangle$.

###### Proof.

This is [42, Lemma 1.4]. ∎

In the next lemma, the containment
$\mathrm{Aut}_{\mathcal{F}}(E)\subseteq\mathcal{G}$ means that $E$ is an
object in $\mathcal{G}$ and that
$\mathrm{Aut}_{\mathcal{F}}(E)=\mathrm{Aut}_{\mathcal{G}}(E)$. We will use
this notation from here on.

######  Lemma 5.13.

Suppose that $\mathcal{F}$ and $\mathcal{G}$ are saturated fusion systems with
${\mathcal{G}}\subseteq\mathcal{F}$. Assume that $E$ is
$\mathcal{F}$-essential and
$\mathrm{Aut}_{\mathcal{F}}(E)\subseteq\mathcal{G}$. Then $E$ is
${\mathcal{G}}$-essential.

###### Proof.

We have $\mathrm{Aut}_{\mathcal{F}}(E)=\mathrm{Aut}_{\mathcal{G}}(E)$ and
$E^{\mathcal{G}}\subseteq E^{\mathcal{F}}$. Therefore, as $E$ is fully
$\mathcal{F}$-normalized and $\mathcal{F}$-centric it is also fully
${\mathcal{G}}$-normalized and ${\mathcal{G}}$-centric. Since
$\mathrm{Out}_{\mathcal{F}}(E)=\mathrm{Out}_{\mathcal{G}}(E)$, we know
$\mathrm{Out}_{\mathcal{G}}(E)$ has a strongly $p$-embedded subgroup. Thus $E$
is $\mathcal{G}$-essential. ∎

Recall that, following [5, Proposition I.3.3], if $P<S$ is fully
$\mathcal{F}$-normalized, $H_{\mathcal{F}}(P)$ is defined to be the subgroup
of $\mathrm{Aut}_{\mathcal{F}}(P)$ which is generated by those morphisms of
$P$ which extend to $\mathcal{F}$-isomorphisms between strictly larger
subgroups of $S$. Then the statement in [5, Proposition I.3.3], includes the
fact that, if $P$ is $\mathcal{F}$-essential, then
$H_{\mathcal{F}}(P)/\mathrm{Inn}(P)$ is strongly $p$-embedded in
$\mathrm{Aut}_{\mathcal{F}}(P)$.

The next result is well-known and we use it to show that certain subgroups are
not $\mathcal{F}$-essential.

######  Lemma 5.14.

Suppose that $\mathcal{F}$ is a saturated fusion system on $S$ and $Q\leq S$.
Assume that $Q_{s}<Q_{s-1}<\dots<Q_{0}=Q$ are
$\mathrm{Aut}_{\mathcal{F}}(Q)$-invariant with $Q_{s}\leq\Phi(Q)$. If
$A\leq\mathrm{Aut}_{S}(Q)$ and $[Q_{i},A]\leq Q_{i+1}$ for $0\leq i\leq s-1$,
then $A\leq O_{p}(\mathrm{Aut}_{\mathcal{F}}(Q))$. In particular, if
$A\not\leq\mathrm{Inn}(Q)$, then $Q$ is not $\mathcal{F}$-essential.

###### Proof.

Set $B=\langle A^{\mathrm{Aut}_{\mathcal{F}}(Q)}\rangle$. Assume that
$\beta\in B$ has order coprime to $p$. Then $[Q,\beta;s]\leq Q_{s}$ and so
[25, Theorem 5.3.6] implies that $[Q,\beta]\leq Q_{s}\leq\Phi(Q)$. Thus [25,
Theorem 5.1.4] implies $\beta=1$ and so $B$ is a $p$-group. Therefore $A\leq
B\leq O_{p}(\mathrm{Aut}_{\mathcal{F}}(Q))$ and this proves the first claim.
Since $O_{p}(\mathrm{Aut}_{\mathcal{F}}(E))=\mathrm{Inn}(E)$ for
$E\in\mathcal{E}_{\mathcal{F}}$, if $A\not\leq\mathrm{Inn}(Q)$, then $Q$ is
not $\mathcal{F}$-essential. ∎

If $Q$ has a series of subgroups as in Lemma 5.14 and $A$ centralizes
$Q_{i}/Q_{i+1}$, then we say that $A$ _stabilizes the series_
$Q_{s}<Q_{s-1}<\dots<Q_{0}=Q$.

The next lemma relies on the fact that we know all non-abelian simple groups
with a strongly $p$-embedded subgroup which contain an elementary abelian
subgroup of order $p^{2}$.

######  Lemma 5.15.

Suppose that $\mathcal{F}$ is a saturated fusion system on $S$ and $E$ is
$\mathcal{F}$-essential. Then $|E/\Phi(E)|\geq|N_{S}(E)/E|^{2}$.

###### Proof.

This is [49, Proposition 4.6 (4)]. ∎

The next lemma relies on work of Henke [31].

######  Proposition 5.16.

Suppose that $\mathcal{F}$ is a saturated fusion system on the $p$-group $S$
and $E$ is an $\mathcal{F}$-essential subgroup. Assume that $U$ and $W$ are
$\mathrm{Aut}_{\mathcal{F}}(E)$-invariant subgroups of $E$ with $V=U/W$
centralized by $\mathrm{Inn}(E)$ and non-trivial as an
$O^{p^{\prime}}(\mathrm{Aut}_{\mathcal{F}}(E))$-module. If
$|[V,\mathrm{Out}_{S}(E)]|\leq p$ or $|V:C_{V}(\mathrm{Out}_{S}(E))|\leq p$,
then

$O^{p^{\prime}}(\mathrm{Aut}_{\mathcal{F}}(E))/C_{O^{p^{\prime}}(\mathrm{Aut}_{\mathcal{F}}(E))}(V)\cong\mathrm{SL}_{2}(p),$

and $V/C_{V}(O^{p^{\prime}}(\mathrm{Aut}_{\mathcal{F}}(E)))$ is the natural
$\mathrm{SL}_{2}(p)$-module.

###### Proof.

The hypothesis that $V$ is a non-trivial
$O^{p^{\prime}}(\mathrm{Aut}_{\mathcal{F}}(E))$-module implies that
$\mathrm{Aut}_{S}(E)$ does not centralize $V$. Hence
$|[V,\mathrm{Out}_{S}(E)]|=p\leq|\mathrm{Out}_{S}(E)/C_{\mathrm{Out}_{S}(E)}(V)|$
or
$|V:C_{V}(\mathrm{Out}_{S}(E))|=p\leq|\mathrm{Out}_{S}(E)/C_{\mathrm{Out}_{S}(E)}(V)|$.
This, by definition, means that either the dual of $V$ or $V$ is a so-called
failure of factorisation module for
$O^{p^{\prime}}(\mathrm{Aut}_{\mathcal{F}}(E))/C_{O^{p^{\prime}}(\mathrm{Aut}_{\mathcal{F}}(E))}(V)$.
Using [31, Theorem 5.6], we obtain
$O^{p^{\prime}}(\mathrm{Aut}_{\mathcal{F}}(E))/C_{O^{p^{\prime}}(\mathrm{Aut}_{\mathcal{F}}(E))}(V)\cong\mathrm{SL}_{2}(p^{a})$
and $V/C_{V}(O^{p^{\prime}}(\mathrm{Aut}_{\mathcal{F}}(E)))$ is the natural
$\mathrm{SL}_{2}(p^{a})$-module. Since one of $|[V,\mathrm{Out}_{S}(E)]|=p$ or
$|V:C_{V}(\mathrm{Out}_{S}(E))|=p$, we deduce that $p=p^{a}$ and this
completes the proof. ∎

We continue this section by recalling important properties of
$\mathcal{F}$-pearls (see Definition 1.1).

######  Lemma 5.17.

Suppose that $p$ is an odd prime, $\mathcal{F}$ is a saturated fusion system
on a $p$-group $S$ and $P\in\mathcal{P}(\mathcal{F})$. Then

  1. (i)

$S$ has maximal class;

  2. (ii)

$\mathrm{Out}_{\mathcal{F}}(P)$ is isomorphic to a subgroup of
$\mathrm{GL}_{2}(p)$ and
$O^{p^{\prime}}(\mathrm{Out}_{\mathcal{F}}(P))\cong\mathrm{SL}_{2}(p)$;

  3. (iii)

$P/\Phi(P)$ is a natural $\mathrm{SL}_{2}(p)$-module for
$O^{p^{\prime}}(\mathrm{Out}_{\mathcal{F}}(P))$;

  4. (iv)

$[{N}_{S}(P)\colon P]=p$;

  5. (v)

every subgroup of $S$ that is $\mathcal{F}$-conjugate to $P$ is an
$\mathcal{F}$-pearl.

###### Proof.

This is a combination of [29, Lemma 1.5, Corollary 1.11 and Lemma 1.13]. ∎

######  Lemma 5.18.

Suppose that $p$ is an odd prime, $\mathcal{F}$ is a saturated fusion system
on a $p$-group $S$ of maximal class with $|S|\geq p^{4}$. If $E\leq S$ is an
$\mathcal{F}$-essential subgroup of $S$, then the following are equivalent:

  1. (i)

$E$ is an $\mathcal{F}$-pearl;

  2. (ii)

$E$ is not contained in $\gamma_{1}(S)$ or in $C_{S}(Z_{2}(S))$; and

  3. (iii)

there exists an element $x\in S\setminus C_{S}(Z_{2}(S))$ of order $p$ such
that either $E=\langle x\rangle Z(S)$ or $E=\langle x\rangle Z_{2}(S)$.

###### Proof.

This is a restatement of [29, Lemma 2.4]. ∎

The next lemma helps when we apply Lemma 5.12.

######  Lemma 5.19.

Suppose that $p$ is an odd prime, $\mathcal{F}$ is a saturated fusion system
on a $p$-group $S$ with $|S|\geq p^{4}$. If $P\in\mathcal{P}(\mathcal{F})$,
then no proper subgroup of $P$ is $\mathcal{F}$-centric.

###### Proof.

There is nothing to do if $P$ is abelian. Assume $P$ is extraspecial. Since no
subgroup of order $p$ is $\mathcal{F}$-centric we can assume $Q\leq P$ has
order $p^{2}$. Then parts (ii) and (iii) of Lemma 5.17 imply that $Q$ is
$\mathrm{Aut}_{\mathcal{F}}(P)$ conjugate to $Z_{2}(S)\leq P$. Since $|S|\geq
p^{4}$, and $|S:C_{S}(Z_{2}(S))|=p$, we conclude $Z_{2}(S)$ and therefore $Q$
are not $\mathcal{F}$-centric. ∎

In the next lemma for a subgroup $A$ of a group $B$, we define $N^{0}(A)=A$
and then, for $i>0$, $N^{i}(A)=N_{B}(N^{i-1}(A))$. The ordered collection of
these subgroups is called the _normalizer tower_ of $A$ in $B$ and its length
is the minimal $\ell$ such that $N^{\ell+1}(A)=N^{\ell}(A)$.

######  Theorem 5.20.

Suppose that $p$ is an odd prime, $\mathcal{F}$ is a saturated fusion system
on a $p$-group $S$ and $P\in\mathcal{P}(\mathcal{F})$ is an
$\mathcal{F}$-pearl with $|S:P|=p^{m}$. Then

  1. (i)

the members of the normalizer tower of $P$

${N}^{0}(P)<{N}^{1}(P)<{N}^{2}(P)<\dots<{N}^{m-1}(P)<{N}^{m}(P)=S$

are the only subgroups of $S$ which contain $P$; moreover
$|{N}^{i}(P)\colon{N}^{i-1}(P)|=p$ for every $1\leq i\leq m$;

  2. (ii)

$P$ is not properly contained in any $\mathcal{F}$-essential subgroup of $S$;

  3. (iii)

every morphism in ${N}_{\mathrm{Aut}_{\mathcal{F}}(P)}(\mathrm{Aut}_{S}(P))$
is the restriction of an automorphism of $S$ that normalizes each
${N}^{i}(P)$.

###### Proof.

This is an application of [29, Theorem 3.6]. ∎

######  Lemma 5.21.

Suppose that $p$ is an odd prime, $\mathcal{F}$ is a saturated fusion system
on a $p$-group $S$. Let $P\in\mathcal{P}(\mathcal{F})$ and let $m$ be such
that $|S:P|=p^{m}$. For every $1\leq i\leq m-1$ let $\mathcal{F}_{i}$ be the
smallest fusion subsystem of $\mathcal{F}$ defined on ${N}^{i}(P)$ such that
$\mathrm{Aut}_{\mathcal{F}_{i}}(P)=\mathrm{Aut}_{\mathcal{F}}(P)$ and
$\mathrm{Aut}_{\mathcal{F}_{i}}({N}^{i}(P))=\mathrm{Inn}({N}^{i}(P)){N}_{\mathrm{Aut}_{\mathcal{F}}({N}^{i}(P))}(P)$.
Then $\mathcal{F}_{i}$ is a saturated fusion subsystem of $\mathcal{F}$ and
$P\in\mathcal{P}(\mathcal{F}_{i})$.

###### Proof.

This is [29, Lemma 3.7]. ∎

######  Lemma 5.22.

Assume that $\mathcal{F}$ is a saturated fusion system on a $p$-group $S$,
$\mathcal{C}$ is a set of $\mathcal{F}$-class representatives of
$\mathcal{F}$-essential subgroups, and $P\in\mathcal{C}$. If $P$ is an
$\mathcal{F}$-pearl, then
$\mathcal{G}=\langle\mathrm{Aut}_{\mathcal{F}}(S),\mathrm{Aut}_{\mathcal{F}}(E)\mid
E\in\mathcal{C}\setminus\\{P\\}\rangle$ is saturated.

###### Proof.

This is [49, Lemma 6.5]. ∎

######  Theorem 5.23.

Suppose that $p$ is an odd prime, $\mathcal{F}$ is a saturated fusion system
on a $p$-group $S$ of order $p^{n}$ with $n\geq 4$. Assume that
$\mathcal{E}_{\mathcal{F}}=\mathcal{P}_{a}(\mathcal{F})$ and $\gamma_{1}(S)$
not abelian or extraspecial. If $n\not\equiv 1\pmod{p-1}$, then $\mathcal{F}$
is simple and exotic.

###### Proof.

Let $P\in\mathcal{E}_{\mathcal{F}}$ and assume that $n\not\equiv 1\pmod{p-1}$.
Let $\mathcal{G}_{0}$ be the saturated fusion system obtained by using Lemma
5.22 to remove $\mathcal{F}$-pearls until
$\mathcal{E}_{\mathcal{G}_{0}}=P^{\mathcal{G}_{0}}$. Then $\mathcal{G}_{0}$ is
simple by [29, Theorem 4.3]. It follows that $\mathcal{F}$ is simple. Suppose
that $R\leq S$ is strongly closed. Then, as $S$ has maximal class by Lemma
5.17 (i), $Z(S)\leq R$ and as $\mathrm{Aut}_{\mathcal{F}}(P)$ acts irreducibly
on $P$, $P\leq R$. Since $R$ is normal in $S$ it follows that $R\geq\langle
P^{S}\rangle=P\gamma_{2}(S)$. Hence $R$ has maximal class and is centric. In
particular, $R$ is centric and does not factor as a product of two or more
subgroups. Hence, if $\mathcal{F}$ is realizable, then [42, Lemma 1.5] shows
that it is realized by a finite simple group. Since $\gamma_{1}(S)$ is neither
abelian nor extraspecial, Lemma 3.7 shows that $\mathcal{F}$ is not
realizable. ∎

The next theorem can be used to add pearls to fusion systems.

######  Theorem 5.24.

Let $\mathcal{F}_{0}$ be a saturated fusion system on a finite $p$-group $S$.
Let $V\leq S$ be a fully $\mathcal{F}_{0}$-normalized subgroup, set
$H=\mathrm{Out}_{\mathcal{F}_{0}}(V)$ and let
$\widetilde{\Delta}\leq\mathrm{Out}(V)$ be such that $H$ is a strongly
$p$-embedded subgroup of $\widetilde{\Delta}$. For $\Delta$ the full preimage
of $\tilde{\Delta}$ in $\mathrm{Aut}(V)$, write

$\mathcal{F}=\langle\mathrm{Mor}(\mathcal{F}_{0}),\Delta\rangle.$

Assume further that

  1. (i)

$V$ is $\mathcal{F}_{0}$-centric and minimal under inclusion amongst all
$\mathcal{F}$-centric subgroups; and

  2. (ii)

no proper subgroup of $V$ is $\mathcal{F}_{0}$-essential.

Then $\mathcal{F}$ is saturated.

###### Proof.

This is a statement of [54, Theorem C] for the special case $m=1$. ∎

The next proposition allows us in Section 13 to exploit results about fusion
systems on $p$-groups with an abelian subgroup of index $p$ [42, 20, 44].

######  Proposition 5.25.

Suppose that $p$ is an odd prime, $\mathcal{F}$ is a saturated fusion system
on $S$, $P\in\mathcal{P}(\mathcal{F})$ and
$\mathrm{Aut}_{\mathcal{F}}(S)=\mathrm{Inn}(S){N}_{\mathrm{Aut}_{\mathcal{F}}(S)}(P)$.
Assume that $Z(S)<V\leq Z(\gamma_{1}(S))$ is
$\mathrm{Aut}_{\mathcal{F}}(S)$-invariant, put $S_{1}=VP$,

$H=\mathrm{Inn}(S_{1})\langle\phi|_{S_{1}}\mid\phi\in
N_{\mathrm{Aut}_{\mathcal{F}}(S)}(P)\rangle\leq\mathrm{Aut}_{\mathcal{F}}(S_{1})$

and

$B=\\{\phi|_{V}\mid\phi\in\mathrm{Aut}_{\mathcal{F}}(\gamma_{1}(S))\\}\leq\mathrm{Aut}_{\mathcal{F}}(V).$

Then the fusion system

$\mathcal{G}=\langle\mathrm{Aut}_{\mathcal{F}}(P),B,H\rangle$

on $S_{1}$ is saturated and $\mathrm{Aut}_{\mathcal{G}}(S_{1})=H$.

###### Proof.

We have $P\leq S_{1}$ and so, if $|S_{1}:P|=p^{k}$, $S_{1}=N^{k}(P)$ by
Theorem 5.20 (i). Let
$\mathcal{F}_{k}=\langle\mathrm{Aut}_{\mathcal{F}}(P),H\rangle$. Then
$\mathcal{F}_{k}$ is the smallest fusion subsystem of $\mathcal{F}$ defined on
$S_{1}$ such that
$\mathrm{Aut}_{\mathcal{F}_{k}}(P)=\mathrm{Aut}_{\mathcal{F}}(P)$ and
$\mathrm{Aut}_{\mathcal{F}_{k}}(S_{1})=\mathrm{Inn}(S_{1})N_{\mathrm{Aut}_{\mathcal{F}}(S_{1})}(P)$.
By Lemma 5.21, $\mathcal{F}_{k}$ is saturated.

Since $V$ is normal in $S_{1}$, $V$ is fully $\mathcal{F}_{k}$-normalized and,
as $V>Z(S)$, $V$ is abelian and $[S_{1}\colon V]=p$, $V$ is an
$\mathcal{F}_{k}$-centric subgroup of $S_{1}$. We have
$\mathrm{Aut}_{\mathcal{F}_{k}}(V)=\mathrm{Aut}_{H}(V)$ is a subgroup of
$B\leq\mathrm{Aut}_{\mathcal{F}}(V)$. If $\mathrm{Aut}_{H}(V)=B$, then
$\mathcal{G}=\mathcal{F}_{k}$ and $\mathcal{G}$ is saturated. So we may assume
that $\mathrm{Aut}_{H}(V)<B$. In particular, as
$\mathrm{Aut}_{\mathcal{F}}(S)=\mathrm{Inn}(S){N}_{\mathrm{Aut}_{\mathcal{F}}(S)}(P)$,
$\mathrm{Aut}_{\mathcal{F}}(\gamma_{1}(S))>\mathrm{Aut}_{\mathrm{Aut}_{\mathcal{F}}(S)}(\gamma_{1}(S))$
and so $\gamma_{1}(S)$ is $\mathcal{F}$-essential and
$\mathrm{Out}_{\mathrm{Aut}_{\mathcal{F}}(S)}(\gamma_{1}(S))$ is strongly
$p$-embedded in $\mathrm{Out}_{\mathcal{F}}(\gamma_{1}(S))$. Therefore
$\mathrm{Out}_{H}(V)$ is strongly $p$-embedded in $\mathrm{Out}_{B}(V)$.
Furthermore, as $V$ is abelian, $V$ is minimal by inclusion amongst all
$\mathcal{F}_{k}$-centric subgroups and no proper subgroup of $V$ is
$\mathcal{F}_{k}$-essential. Now application of Theorem 5.24 delivers
$\mathcal{G}$ is saturated. That $\mathrm{Aut}_{\mathcal{G}}(S_{1})=H$ is
apparent from its construction. ∎

##  6\. Fusion systems on groups of maximal class: generalities

This section begins the study of saturated fusion systems on maximal class
$p$-groups.

######  Lemma 6.1.

Suppose that $S$ is a $2$-group of maximal class and $\mathcal{F}$ is a
saturated fusion system on $S$. Then $\mathcal{F}$ is known. In particular, if
$E$ is an $\mathcal{F}$-essential subgroup of $S$, then $E$ is an
$\mathcal{F}$-pearl and $\gamma_{1}(S)$ is abelian.

###### Proof.

If $|S|=8$, then there is nothing to do as $\mathcal{F}$-essential subgroups
have order at least $p^{2}$. When $|S|\geq 16$, see [5, Example I.3.8]. ∎

######  Lemma 6.2.

Suppose that $S$ is a $3$-group of maximal class and $\mathcal{F}$ is a
saturated fusion system on $S$. Then $\mathcal{F}$ is known. Furthermore, if
$E$ is an $\mathcal{F}$-essential subgroup of $S$, then $E$ is either an
$\mathcal{F}$-pearl or $E=\gamma_{1}(S)$ and in this case $\gamma_{1}(S)$ is
abelian.

###### Proof.

Note that by Lemma 3.1(iv), $|\Omega_{1}(S)|\leq 3^{3}$, with equality only if
$|S|=3^{4}$ and $\gamma_{1}(S)$ is elementary abelian (of index $3$ in $S$).
Therefore the classification of the fusion systems on these groups can be read
from [21, 48, 42, 20]. Inspection of the theorems listed gives the statement.
∎

Because of Lemmas 6.1 and 6.2, for the remainder of this article we focus on
the case $p\geq 5$.

######  Hypothesis 6.3.

$p$ is a prime at least $5$ and $\mathcal{F}$ is a saturated fusion system on
a maximal class $p$-group $S$ of order at least $p^{4}$.

We start with some lemmas which loosely locate the potential
$\mathcal{F}$-essential subgroups within $S$.

######  Lemma 6.4.

If $E$ is an $\mathcal{F}$-essential subgroup which is not an
$\mathcal{F}$-pearl, then either $E\leq\gamma_{1}(S)$ or
$E\leq{C}_{S}({Z}_{2}(S))$. Furthermore, there are no $\mathcal{F}$-pearls in
$\gamma_{1}(S)$ or in ${C}_{S}({Z}_{2}(S))$.

###### Proof.

The first part of the statement is Lemma 5.18. Suppose $E$ is an
$\mathcal{F}$-pearl. We aim to show that $E$ is not contained in
$\gamma_{1}(S)$ or in ${C}_{S}({Z}_{2}(S))$.

If $E\leq{C}_{S}({Z}_{2}(S))$, then
${Z}_{2}(S)\leq{Z}({C}_{S}({Z}_{2}(S)))\leq Z(E)\leq E$ because $E$ is
$\mathcal{F}$-centric. Since $|Z_{2}(S)|=p^{2}$ and
$E\leq{C}_{S}({Z}_{2}(S))$, we deduce that $E=C_{S}(Z_{2}(S))=Z_{2}(S)$ is
abelian, which is impossible as $|S|\geq p^{4}$ and $|S:C_{S}(Z_{2}(S))|=p$.
Therefore $E\nleq{C}_{S}({Z}_{2}(S))$. In particular, we can assume
$\gamma_{1}(S)\neq C_{S}(Z_{2}(S))$, $S$ is exceptional and
$E\leq\gamma_{1}(S)$. Thus, as $S$ is exceptional, $|S|\geq p^{6}$ by Lemma
3.2 (v). Note that, as $E\leq\gamma_{1}(S)$, $[{Z}_{3}(S),E]\leq{Z}(S)<E$, so
${Z}_{3}(S)\leq{N}_{S}(E)$ and $|E|\geq p^{3}$. Also
$E\nleq{Z}_{3}(S)\leq{C}_{S}({Z}_{2}(S))$. Since $[{N}_{S}(E)\colon E]=p$, we
conclude that ${N}_{S}(E)={Z}_{3}(S)E$ and $E$ is extraspecial. However, the
series $1<{Z}(S)={Z}(E)<E$ is stabilized by ${Z}_{3}(S)$, and therefore
${Z}_{3}(S)\leq O_{p}(\mathrm{Aut}(E))$ by Lemma 5.14 which is a
contradiction. Hence, if $E$ is an $\mathcal{F}$-pearl, then $E$ is not
contained in either in $\gamma_{1}(S)$ or in ${C}_{S}({Z}_{2}(S))$. ∎

The next result is included for completeness.

######  Corollary 6.5.

If $\gamma_{1}(S)$ is abelian, then the candidates for $\mathcal{F}$-essential
subgroups are $\gamma_{1}(S)$ and $\mathcal{F}$-pearls. In particular, if
$O_{p}(\mathcal{F})=1$, then $\mathcal{F}$ has an $\mathcal{F}$-pearl.

###### Proof.

(See also [42, Lemma 2.3 (a)]). Since $\gamma_{1}(S)$ is abelian, $S$ is not
exceptional. Thus, if $E$ is $\mathcal{F}$-essential and not an
$\mathcal{F}$-pearl, then $E\leq\gamma_{1}(S)$ by Lemma 6.4. As $E$ is
$\mathcal{F}$-centric, this yields $E=\gamma_{1}(S)$ as claimed. In
particular, if $\mathcal{F}$ has no $\mathcal{F}$-pearls, then $\gamma_{1}(S)$
is the only $\mathcal{F}$-essential subgroup in $\mathcal{F}$. Since
$\gamma_{1}(S)$ is a characteristic subgroup of $S$, the Alperin-Goldschmidt
Theorem yields $\gamma_{1}(S)=O_{p}(\mathcal{F})$. Hence, if
$O_{p}(\mathcal{F})\neq 1$, then $\mathcal{F}$ has an $\mathcal{F}$-pearl. ∎

######  Proposition 6.6.

Suppose $E$ is $\mathcal{F}$-essential and $E$ is normal in $S$. Then $E$ is a
maximal subgroup of $S$. Moreover, either
$E\in\\{\gamma_{1}(S),{C}_{S}({Z}_{2}(S))\\}$ or $|S|=p^{4}$ and $E$ is a non-
abelian $\mathcal{F}$-pearl.

###### Proof.

Suppose for a contradiction that $E$ is not a maximal subgroup of $S$. Since
$E\mathrel{\unlhd}S$ and $S$ has maximal nilpotency class, $E$ is a member of
the lower central series of $S$. Hence $E=\gamma_{k}(S)$ for some $k\geq 2$.
In particular, $\mathrm{Out}_{S}(E)=S/E$ is a group of maximal class. Since
$p$ is odd, $S/E$ contains an elementary abelian subgroup of order $p^{2}$ by
[25, 5.4.10 (ii)]. Lemma 2.10 yields that
$F^{*}(\mathrm{Out}_{\mathcal{F}}(E)/O_{p^{\prime}}(\mathrm{Out}_{\mathcal{F}}(E)))$
is a non-abelian simple group. Set
$W=F^{*}(\mathrm{Out}_{\mathcal{F}}(E)/O_{p^{\prime}}(\mathrm{Out}_{\mathcal{F}}(E)))$
and
$X=W\mathrm{Out}_{S}(E)O_{p^{\prime}}(\mathrm{Out}_{\mathcal{F}}(E))/O_{p^{\prime}}(\mathrm{Out}_{\mathcal{F}}(E))$.
Using Theorem 2.12 we know the candidates for $X$. Remember also that $p\geq
5$. Suppose first that $\mathrm{Out}_{S}(E)$ is abelian. Then
$\mathrm{Out}_{S}(E)\cong S/S^{\prime}$ is elementary abelian of order
$p^{2}$. The possibilities for $X$ are $\mathrm{PSL}_{2}(p^{2})$,
$\mathrm{Alt}(2p)$ for arbitrary odd $p$, and
${}^{2}\mathrm{F}_{4}(2)^{\prime}$ or $\mathrm{Fi}_{22}$ with $p=5$. By a
Frattini Argument and using [28, Theorem 7.8.1],
${N}_{\mathrm{Aut}_{\mathcal{F}}(E)}(\mathrm{Aut}_{S}(E))$ acts irreducibly on
$\mathrm{Out}_{S}(E)$. As every morphism in
${N}_{\mathrm{Aut}_{\mathcal{F}}(E)}(\mathrm{Aut}_{S}(E))$ is the restriction
of an $\mathcal{F}$-automorphism of $S$ (because $E$ is
$\mathcal{F}$-receptive), there exists an automorphism
$\tau\in\mathrm{Aut}_{\mathcal{F}}(S)$ such that
$\gamma_{1}(S)\tau\neq\gamma_{1}(S)$, which is absurd as this subgroup is
characteristic in $S$. If $\mathrm{Out}_{S}(E)$ is non-abelian, then
$\mathrm{Out}_{S}(E)$ is extraspecial and the possibilities for $X$ are
$\mathrm{PSU}_{3}(p)$ for all $p\geq 5$, $\mathrm{McL}$ or
${}^{2}\mathrm{B}_{2}(32){:}5$ with $p=5$, and $\mathrm{J}_{4}$ with $p=11$.
This time [28, Theorem 7.6.2] shows that either
${N}_{\mathrm{Aut}_{\mathcal{F}}(E)}(\mathrm{Aut}_{S}(E))$ acts irreducibly on
$\mathrm{Out}_{S}(E)/Z(\mathrm{Out}_{S}(E))$ or
$X\cong{}^{2}\mathrm{B}_{2}(32){:}5$ and $\mathrm{Out}_{S}(E)\cong
5^{1+2}_{-}$ respectively. In the former case, we obtain a contradiction
exactly as in the abelian case. Suppose that
$X\cong{}^{2}\mathrm{B}_{2}(32){:}5$. Then $\mathrm{Out}_{S}(E)\cong
S/\gamma_{3}(S)$ has exponent 25. Since $|S|\geq 5^{4}$, applying Lemma 3.1
(vi) to $S/\gamma_{4}(S)$ yields $S/\gamma_{3}(S)$ has exponent $5$, which is
a contradiction. Therefore, if $E$ is normal in $S$, then $E$ is a maximal
subgroup of $S$.

If $E\not\in\\{\gamma_{1}(S),{C}_{S}({Z}_{2}(S))\\}$, then Theorem 5.18
implies $E$ is an $\mathcal{F}$-pearl and so $|S|=p^{4}$. This proves the
lemma. ∎

######  Lemma 6.7.

Suppose that $|S|\geq p^{5}$ and $E$ is an $\mathcal{F}$-essential subgroup
which is not an $\mathcal{F}$-pearl. Then $|E|\geq p^{4}$.

###### Proof.

Suppose that $|E|\leq p^{3}$. Then, as $E$ is $\mathcal{F}$-essential and $E$
is not an $\mathcal{F}$-pearl, $|E|=p^{3}$ and $E$ is abelian. Furthermore,
Lemma 6.4 indicates that either $E\leq\gamma_{1}(S)$ or $E\leq