Morita equivalence classes of blocks with extraspecial defect groups
$p_{+}^{1+2}$ 111This research was supported by the Marsden Fund (of New
Zealand), via award numbers UoA 1626 and UoA 2030, and the EPSRC (grant no.
EP/T004606/1)

Jianbei An222Department of Mathematics, University of Auckland, Auckland, New
Zealand. Email<EMAIL_ADDRESS>and Charles W. Eaton333Department of
Mathematics, University of Manchester, Manchester M13 9PL, United Kingdom.
Email<EMAIL_ADDRESS>
###### Abstract

We characterise the Morita equivalence classes of blocks with extraspecial
defect groups $p_{+}^{1+2}$ for $p\geq 5$, and so show that Donovan’s
conjecture and the Alperin-McKay conjecture hold for such $p$-groups. For
$p=3$ we reduce Donovan’s conjecture for blocks with defect group
$3_{+}^{1+2}$ to bounding the Cartan invariants for such blocks of quasisimple
groups. We apply the characterisation to the case $p=5$ as an example, to list
the Morita equivalence classes of such blocks.

##  1 Introduction and main results

Let $p$ be a prime and $(K,{\mathcal{O}},k)$ be a modular system with $k$ an
algebraically closed field of characteristic $p$ (we will discuss the choice
of ${\mathcal{O}}$ further later). Donovan’s conjecture, which may be stated
over ${\mathcal{O}}$ or $k$, predicts that for a given finite $p$-group $P$,
there are only finitely many Morita equivalence classes of blocks of finite
groups with defect groups isomorphic to $P$. Whilst significant progress on
the conjecture has been achieved for the prime two, especially for abelian
defect groups, at present for odd primes the only cases for which the
conjecture is known are the cyclic $p$-groups and those $p$-groups admitting
no fusion system other than that of the $p$-group itself (see [20] for a
recent summary, and also [17] where progress on the conjecture is recorded).
We apply the classification of finite simple groups to give a description of
the Morita equivalence classes of ${\mathcal{O}}$-blocks with extraspecial
defect groups of order $p^{3}$ and exponent $p$ when $p\geq 5$, from which it
follows that Donovan’s conjecture holds for these $p$-groups, and further one
may derive a classification of the Morita equivalence classes. As an
illustration, we give the explicit classification of all Morita equivalence
classes of blocks with defect group $5_{+}^{1+2}$. A further application is
that the Alperin-McKay conjecture holds for blocks with defect group
$p_{+}^{1+2}$ for $p\geq 5$.

In the following note that a Morita equivalence is called basic if it is
induced by an endopermutation source bimodule. Our main result is:

###### Theorem 1.1

Let $p\geq 5$ be a prime. Let $G$ be a finite group and $B$ a block of
${\mathcal{O}}G$ with defect group $D\cong p_{+}^{1+2}$. Then $B$ is basic
Morita equivalent to a block of maximal defect of one of the following:

  1. (a)

$D\rtimes\hat{E}$, where $\hat{E}$ is a $p^{\prime}$-central extension of a
$p^{\prime}$-subgroup $E$ of $GL_{2}(p)$ acting faithfully on $D$;

  2. (b)

$(C_{p}\times C_{p})\rtimes H$, where $SL_{2}(p)\leq H\leq GL_{2}(p)$;

  3. (c)

$H$ where $PSL_{3}(p)\leq H/Z(H)\leq{\rm Aut}(PSL_{3}(p))$ and $Z(H)\leq
H^{\prime}$;

  4. (d)

$H$ where $PSU_{3}(p)\leq H/Z(H)\leq{\rm Aut}(PSU_{3}(p))$ and $Z(H)\leq
H^{\prime}$;

  5. (e)

$2.HS$, $2.HS.2$, $3.McL$, $McL.2$, $2.Ru$, $Co_{3}$, $Co_{2}$ or $Th$, where
$p=5$;

  6. (f)

$He$, $He.2$, $3.O^{\prime}N$, $O^{\prime}N.2$, $3.Fi_{24}^{\prime}$ or
$Fi_{24}^{\prime}.2$, where $p=7$;

  7. (g)

$J_{4}$, where $p=11$;

  8. (h)

the Fischer-Griess Monster $M$, where $p=13$.

Further, the Morita equivalence preserves both the fusion of the block and the
Külshammer-Puig class of the blocks.

###### Corollary 1.2

Donovan’s conjecture holds for ${\mathcal{O}}$-blocks for $p_{+}^{1+2}$ for
primes $p\geq 5$. That is, for each $p\geq 5$ there are only finitely many
Morita equivalence classes amongst ${\mathcal{O}}$-blocks of finite groups
with defect group $p_{+}^{1+2}$.

Numerical invariants for blocks with these defect groups have been studied in
[28], although they are not completely determined, and in [29] it is shown
that Olsson’s conjecture holds for these blocks, i.e., that the number of
irreducible characters of height zero is at most $[D:D^{\prime}]=p^{2}$.
Principal blocks with such defect groups have been studied in [35] using the
classification of finite simple groups, where the existence of a generalised
perfect isometry is shown. Some of the methods in that paper motivate those
used here.

A consequence of Theorem 1.1 is that the Alperin-McKay conjecture holds in
this case. Here $k(B)$ denotes the number of irreducible character in $B$. We
write $k_{h}(B)$ for the number of irreducible characters of height $h$, where
the height of an irreducible character $\chi$ in a block with defect group $D$
is the non-negative integer $h$ such that $\chi(1)_{p}=p^{h}[G:D]_{p}$.

###### Corollary 1.3

Let $B$ be a block of a finite group with defect group $D\cong p_{+}^{1+2}$
for $p\geq 5$. Let $b$ be the Brauer correspondent of $B$ in $N_{G}(D)$. Then
$k_{0}(B)=k_{0}(b)$.

Two important components of the proof of Theorem 1.1 are the following: the
classification of blocks of quasisimple groups with extraspecial defect groups
in [5]; an analysis of blocks of quasisimple groups with defect group
$C_{p}\times C_{p}$ and a non-inner automorphism of order $p$, making use of
[3].

We also obtain detailed information about blocks with defect group
$3_{+}^{1+2}$, found in Proposition 6.2. However, there are infinite classes
of blocks with such defect groups within which Morita equivalence has yet to
be decided, so we cannot at present extend Theorem 1.1 to $p=3$. However, we
can make the following reduction for Donovan’s conjecture. For a block $B$ let
${\rm c}(B)$ be the largest entry of its Cartan matrix.

###### Theorem 1.4

Suppose that there is $c\in\mathbb{N}$ such that ${\rm c}(B)\leq c$ for all
quasisimple groups $G$ and all blocks $B$ of $kG$ with defect group
$3_{+}^{1+2}$. Then Donovan’s conjecture holds for ${\mathcal{O}}$-blocks with
defect group $3_{+}^{1+2}$.

Note that the fact that we need only examine Cartan invariants for these
groups rather than Morita equivalence classes stems from Farrell and Kessar’s
determination of Morita-Frobenius numbers of blocks of quasisimple groups in
[22].

As an example of the application of Theorem 1.1, we give the complete
classification of the sixty-two Morita equivalence classes of blocks with
defect group $5_{+}^{1+2}$. Similar classifications could be produced for the
remaining primes $p\geq 5$, although the determination of precise Morita
equivalence classes for the blocks with normal defect groups would require ad
hoc results in each case.

We do not study the groups $p_{-}^{1+2}$ of exponent $p^{2}$ in this article.
Whilst many of the reductions for blocks with these defect groups are the same
as for the exponent $p$ case, a major difference is that it would require the
examination of blocks of quasisimple groups with cyclic defect groups of order
$p^{2}$, which would involve a very different analysis.

The structure of the paper is as follows. In Section 2 we give some basic
notation and prove some elementary general results that we will need later. We
introduce notation necessary for the analysis of classical groups in Section
3. The analysis of blocks of quasisimple groups with defect group $C_{p}\times
C_{p}$ and a non-inner automorphism of order $p$ is contained in Sections 4
and 5. We prove Theorem 1.1 and Theorem 1.4 in Section 6. We analyse the case
$p=5$ in full detail in Section 7. In Section 8 we prove Corollary 1.3.

##  2 Notation and background results

Let $p$ be a prime and let ${\mathcal{O}}$ be a complete discrete valuation
ring with algebraically closed residue field $k$ of characteristic $p$, so
that $k=\mathcal{O}/J(\mathcal{O})$. Let $\mathcal{K}$ be the field of
fractions for ${\mathcal{O}}$. The results of this paper hold in this
generality, however for a canonical statement of Donovan’s conjecture one may
take $k$ to be the algebraic closure of the field $\mathbb{F}_{p}$ of $p$
elements and ${\mathcal{O}}$ to be the ring of Witt vectors for $k$.

For $G$ a finite group, write ${\rm Blk}(G)$ for the set of blocks of
$\mathcal{O}G$. Denote by ${\rm Irr}(B)$ the set of irreducible
$\mathcal{K}$-characters belonging to a block $B\in{\rm Blk}(G)$. We will
often use $D=D(B)$ to denote a defect group of $B$. Denote by ${\rm IBr}(B)$
for the set of irreducible Brauer characters in $B$. Write $k(B)=|{\rm
Irr}(B)|$ and $l(B)=|{\rm IBr}(B)|$. Write $B_{0}(G)$ for the principal block
of ${\mathcal{O}}G$.

A $B$-subgroup is a pair $(Q,b_{Q})$, where $Q$ is a $p$-subgroup of $G$ and
$b_{Q}$ is a block of $C_{G}(Q)$ with Brauer correspondent $(b_{Q})^{G}=B$.
The $B$-subgroups with $|Q|$ maximized are called the Sylow $B$-subgroups, and
they are the $B$-subgroups for which $Q$ is a defect group for $B$. We denote
by $N_{G}(D,b_{D})$ the stabilizer in $N_{G}(D)$ of $(D,b_{D})$ under
conjugation.

The _inertial quotient_ of $B$ is $E=N_{G}(D,b_{D})/DC_{G}(D)$, together with
the action of $E$ on $D$. The inertial quotient is determined by the fusion
system $\mathcal{F}=\mathcal{F}_{(D,b_{D})}(G,B)$ for $B$, sometimes called
the Frobenius category. We refer to [31, Section 8.5] for background on this.
Following the presentation in [31, Section 8.14], a Külshammer-Puig class is
an element of $H^{2}({\rm Aut}_{\mathcal{F}}(Q))$, which is isomorphic to
$H^{2}(E,k^{\times})$ (see [31, Remark 8.14.3]).

The Morita equivalences occurring paper are all basic Morita equivalences or
source algebra equivalences (a basic Morita equivalence is one induced by an
endopermutation source bimodule, for which see [31, Section 9.10]).

Now suppose $D\lhd G$ and let $E$ be the inertial quotient of $B$. By [31,
Theorem 6.14.1], $B$ is basic Morita equivalent to a twisted group algebra
$\mathcal{O}_{\alpha}(D\rtimes E)$, where $\alpha\in H^{2}(E,k^{\times})$. In
turn, this is basic Morita equivalent to a block of
${\mathcal{O}}(D\rtimes\hat{E})$, where $\hat{E}$ is a central extension of
$E$ by $Z\leq Z(\hat{E})\cap\hat{E}^{\prime}$ with $E$ acting faithfully and
$Z$ acting trivially on $D$. Blocks of ${\mathcal{O}}(D\rtimes\hat{E})$
correspond to irreducible characters of $Z$. We remark that the Morita
equivalence classes are independent of choice of isoclinism class of central
extension. Whilst choice of $E$ and $\alpha$ determines the Morita equivalence
class, it is not always the case that different choices give distinct Morita
equivalence classes. It may (and does) happen that distinct choices of
$\alpha$ give the same Morita equivalence class, as we will see in Section 7.

The outer automorphism group of $p_{+}^{1+2}$ is isomorphic to $GL_{2}(p)$,
and the possible inertial quotients correspond to the conjugacy classes of
subgroups of $GL_{2}(p)$ of order prime to $p$.

We now gather together some elementary results that will be needed later on.

A main task in this paper is to show that if a quasisimple group of Lie type
in characteristic different to $p$ has a block with defect group $C_{p}\times
C_{p}$, then there cannot be a field automorphism of order $p$. The reason for
this is roughly that if the field of definition of our group has exponent
divisible by $p$, then such small but nontrivial defect groups as these cannot
be present. The next lemma is important in showing this. Throughout, we denote
by $m_{p}$ the $p$-part of an integer $m$.

Suppose $p$ is odd. Let $r$ be a prime distinct from $p$ and $q=r^{f}$ for
some integer $f\geq 1$. Let $\epsilon\in\\{+1,-1\\}$ and let $e$ be the
multiplicative order of $\epsilon q$ modulo $p$, so that
$p\mid(q^{e}-\epsilon)$. Write $(q^{e}-\epsilon)_{p}=p^{a}$, where $a\geq 1$
is an integer.

###### Lemma 2.1

With the above notation, if $a=1$, then $p\nmid f$.

Proof: Suppose first that $\epsilon=1$. Let $e^{\prime}$ be the multiplicative
order of $r$ modulo $p$. Calculating in $C_{p}$, we have
$q^{e^{\prime}}=r^{fe^{\prime}}=1$, so $e\mid e^{\prime}$ and
$e^{\prime}=ee_{1}$ for some $e_{1}\geq 1$. But $r^{ef}=1$, so
$e^{\prime}\mid(ef)$ and write $ef=e^{\prime}f_{1}$. Thus

$ef=e^{\prime}f_{1}=ee_{1}f_{1}$

and $f=e_{1}f_{1}$. Let $r_{1}=r^{e_{1}}$, so that $|r_{1}|=e$. If
$p^{b}=(r_{1}^{e}-1)_{p}$ with $b\geq 1$, then $r_{1}^{e}=p^{b}m+1$ for some
integer $m\geq 1$ with $\gcd(m,p)=1$. If $p\mid f$, then $p\mid f_{1}$ and
$f_{1}=pf_{2}$ as $e_{1}\leq e^{\prime}\leq p-1$. Thus
$q^{e}=(r_{1}^{e})^{f_{1}}=((p^{b}m+1)^{f_{2}})^{p}$. In particular,
$p^{b+1}\mid(q^{e}-1)$ and hence $a\geq b+1\geq 2$, a contradiction.

Suppose $\epsilon=-1$. Then $p\mid|U(e,q)|$ with $e$ minimal. If $x\in U(e,q)$
has order $p$, then $Z(U(e,q))=C_{q+1}\leq C_{U(e,q)}(x)=U(1,q^{e})$ and so
$e$ is odd. In particular, $q^{e}\equiv-1\pmod{p}$ and $p\mid(q^{e}+1)$. Since
$p$ is odd, we have $p\nmid(q^{e}-1)$ and $(q^{2e}-1)_{p}=(q^{e}+1)_{p}$. So
if $a=1$, then $(q^{2e}-1)_{p}=p$ and as shown above, $p\nmid 2f$ and hence
$p\nmid f$. $\Box$

###### Lemma 2.2

Let $H\leq GL_{2}(p)$ such that $H$ contains a Sylow $p$-subgroup $P$ of
$GL_{2}(p)$. Then $P\lhd H$ or $SL_{2}(p)\leq H\leq GL_{2}(p)$.

Proof: This is well known and follows easily from [30, Chap. II, 8.27]. $\Box$

###### Lemma 2.3

Let $G$ be a finite group with $C_{G}(O_{p}(G))\leq O_{p}(G)\cong C_{p}\times
C_{p}$ and $|G|_{p}=p^{3}$, where $p$ is an odd prime. Then $G=O_{p}(G)\rtimes
H$ for some $SL_{2}(p)\leq H\leq GL_{2}(p)$.

Proof: We may embed $G/O_{p}(G)$ into ${\rm Aut}(O_{p}(G))\cong GL_{2}(p)$.
Let $P\in{\rm Syl}_{p}(G)$. Since $O_{p}(G)\neq P$, we must have that
$G/O_{p}(G)$ is isomorphic to a subgroup of $GL_{2}(p)$ of order divisible by
$p$ that does not normalise a Sylow $p$-subgroup. Hence by Lemma 2.2
$SL_{2}(p)\leq G/O_{p}(G)$ and it suffices to show that $G$ must be a split
extension. This follows from [14, Theorem 1] by observing that $Z(SL_{2}(p))$
fixes only the zero element of the natural module. $\Box$

##  3 Notation for classical groups

Let $V$ be a linear, unitary, non-degenerate orthogonal or symplectic space
over the field $\mathbb{F}_{q}$, where $q=r^{f}$ with $\gcd(r,p)=1$. We will
follow the notation of [1], [11], [23] and [24].

If $V$ is orthogonal, then there is a choice of equivalence classes of
quadratic forms. Write $\eta(V)$ for the type of $V$ as defined in [24], so
$\eta(V)=+$ or $-$. In addition, let $\eta(V)=+$ or $-$ according as $V$ is
linear or unitary. If $V$ is non-degenerate orthogonal or symplectic, then
denote by $I(V)$ the group of isometries on $V$ and let $I_{0}(V)=I(V)\cap{\rm
SL}(V)$.

If $V$ is symplectic, then $I(V)=I_{0}(V)={{\rm Sp}}_{2n}(q)$.

If $V$ is a $(2n+1)$-dimensional orthogonal space, then
$I(V)=\langle-1_{V}\rangle\times I_{0}(V)$ with $I_{0}(V)={\rm SO}_{2n+1}(q)$.

If $V$ is a $2n$-dimensional orthogonal space, then
$I(V)=O^{\eta}(V)=O^{\eta}_{2n}(q)$ and $I_{0}(V)={\rm SO}^{\eta}_{2n}(q)$.

If $V$ is a $2n$-dimensional non-degenerate orthogonal or symplectic space,
then denote by $J_{0}(V)$ the conformal isometries of $V$ with square
determinant. If $V$ is orthogonal of dimensional at least two, then write
$D_{0}(V)$ for the special Clifford group of $V$ (cf. [24]).

Denote by ${\rm GL}^{+}(V)$ the general linear group ${\rm GL}(V)$ and ${\rm
GL}^{-}(V)$ the unitary group $U(V)$.

Let $G={\rm GL}^{\eta}(V)$ or $I(V)$. Write
$\mathcal{F}_{q}=\mathcal{F}_{q}(G)$ for the set of (monic) polynomials
serving as elementary divisors for semisimple elements of $G$ (cf. [1, p.6]).
If $G={\rm GL}^{-}(V)$, then partition $\mathcal{F}_{q}$ as in [23] by
writing:

$\begin{array}[]{l}\mathcal{F}_{1}=\\{\Delta\in\mathcal{F}_{q}:\Delta\ {\rm
irreducible},\ \Delta\neq X,\ \Delta={\tilde{\Delta}}\\},\\\
\mathcal{F}_{2}=\\{\Delta{\tilde{\Delta}}\in\mathcal{F}_{q}:\Delta\ {\rm
irreducible},\ \Delta\neq X,\ \Delta\neq{\tilde{\Delta}}\\},\end{array}$

where ${\tilde{\Delta}}$ is the monic irreducible polynomial whose roots are
the $q$-th power of those of $\Delta$. If $G=I(V)$, then partition
$\mathcal{F}_{q}$ as in [24] by writing:

$\begin{array}[]{l}\mathcal{F}_{0}=\\{X\pm 1\\},\\\
\mathcal{F}_{1}=\\{\Delta\in\mathcal{F}_{q}\setminus\mathcal{F}_{0}:\Delta\
{\rm irreducible},\ \Gamma\neq X,\ \Delta=\Delta^{*}\\},\\\
\mathcal{F}_{2}=\\{\Delta\Delta^{*}\in\mathcal{F}_{q}\setminus\mathcal{F}_{0}:\Delta\
{\rm irreducible},\ \Delta\neq X,\ \Delta\neq\Delta^{*}\\},\end{array}$

where $\Delta^{*}$ is the monic irreducible polynomial whose roots are the
inverses of those of $\Delta$. Let $d_{\Gamma}$ be the degree of
$\Gamma\in\mathcal{F}_{q}$, and define the reduced degree $\delta_{\Gamma}$ as
in [1] and [24], so that $\delta_{\Gamma}=d_{\Gamma}$ if $G={\rm
GL}^{\eta}(V)$ or $\Gamma\in\mathcal{F}_{0}$, and
$\delta_{\Gamma}=\frac{1}{2}d_{\Gamma}$ if $G=I(V)$ and
$\Gamma\in\mathcal{F}_{1}\cup\mathcal{F}_{2}$.

For each $\Gamma\in\mathcal{F}_{q}$, define $\epsilon_{\Gamma}=1$ when $G={\rm
GL}(V)$, otherwise define

$\epsilon_{\Gamma}=\left\\{\begin{array}[]{cll}\epsilon&if&\Gamma\in\mathcal{F}_{0},\\\
-1&if&\Gamma\in\mathcal{F}_{1},\\\
1&if&\Gamma\in\mathcal{F}_{2}.\end{array}\right.$

Let $e_{\Gamma}$ be the multiplicative order of
$\epsilon_{\Gamma}q^{\delta_{\Gamma}}$ modulo $p$. Thus we may write
$e_{\Gamma}\delta_{\Gamma}=ep^{\alpha_{\Gamma}}\delta_{\Gamma}^{\prime}$ for
some $\alpha_{\Gamma}$ and $\delta_{\Gamma}^{\prime}$ with
$p\nmid\delta_{\Gamma}^{\prime}$.

Let $\mathcal{F}_{q}^{p^{\prime}}=\mathcal{F}_{q}^{p^{\prime}}(G)$ be the
subset of $\mathcal{F}_{q}$ serving as elementary divisors for all semisimple
$p^{\prime}$-elements of $G$.

Given a semisimple element $s\in G$, there is a unique orthogonal
decomposition $V=\sum_{\Gamma\in\mathcal{F}_{q}}V_{\Gamma}(s)$, with
$s=\prod_{\Gamma\in\mathcal{F}_{q}}s_{\Gamma}$, where the $V_{\Gamma}(s)$ are
nondegenerate subspaces of $V$ and $s_{\Gamma}\in U(V_{\Gamma}(s))$, ${\rm
GL}(V_{\Gamma}(s))$ or $I(V_{\Gamma}(s))$ (depending on $G$) has minimal
polynomial $\Gamma$. (For simplicity, we write a direct sum as orthogonal if
$G={\rm GL}(V)$). The decomposition is called the primary decomposition of
$s$. If $s=s_{\Gamma}$, then $s$ is called primitive. Write $m_{\Gamma}(s)$
for the multiplicity of $\Gamma$ in $s_{\Gamma}$. We have
$C_{G}(s)=\prod_{\Gamma\in\mathcal{F}_{q}}C_{\Gamma}(s)$, where
$C_{\Gamma}(s)=I(V_{\Gamma}(s))$ or ${\rm
GL}^{\epsilon_{\Gamma}}(m_{\Gamma}(s),q^{\delta_{\Gamma}})$ as appropriate.

##  4 Classical groups

In this and the following section we analyse blocks of quasisimple groups in
non-defining characteristic with defect group $C_{p}$ or $C_{p}\times C_{p}$.
We show that if such a block exists, then there cannot be a field automorphism
of order $p$. This will help to rule out the existence of blocks of
automorphism groups of such groups with extraspecial defect groups.

Suppose $\eta\in\\{+1,-1\\}$ and $G={\rm GL}_{n}^{\eta}(q)={\rm
GL}^{\eta}(V)$, and let $B_{G}$ be a $p$-block of $G$ with a defect group
$D_{G}$ and label $(s,\kappa)$. Then

$V=V_{0}\perp V_{+},\quad D_{G}=D_{0}\times D_{+},\quad s=s_{0}\times s_{+},$
$None$

where $V_{0}=C_{V}(D_{G})$, $V_{+}=[D_{G},V]$, $s_{0}\in G_{0}={\rm
GL}^{\eta}(V_{0})$ and $s_{+}\in G_{+}:={\rm GL}^{\epsilon}(V_{+})$,
$D_{0}=\langle 1_{V_{0}}\rangle$ and $D_{+}\leq G_{+}$. We also denote ${\rm
GL}^{\eta}(V)$ by $G(V)$ and ${\rm SL}^{\eta}(V)$ by $S(V)$.

###### Proposition 4.1

Let $K_{u}={\rm SL}^{\eta}(V)={\rm SL}^{\eta}_{n}(q)$, $G={\rm GL}^{\eta}(V)$,
$K=K_{u}/Z$ for some $Z\leq Z(K_{u})$, $B\in{\rm Blk}(K)$ with Sylow
$B$-subgroup $(D,b_{K})$. Take $B_{u}\in{\rm Blk}(K_{u})$ dominating $B$,
$B_{G}\in{\rm Blk}(G)$ covering $B_{u}$, $D_{u}=D(B_{u})$ such that
$D_{u}Z/Z=D$ and $D_{G}=D(B_{G})$ such that $D_{u}=D_{G}\cap K_{u}$. Suppose
that $D\cong C_{p}$ or $C_{p}\times C_{p}$. Then either any block of $G/Z$
covering $B$ has a cyclic defect group, or $a=1$ and hence $K$ has no field
automorphism of order $p$.

Proof: Let $(s,\kappa)$ be the label of $B_{G}$, and $V,D_{G},s$ have the
corresponding decomposition (4.1). Let $s_{+}=\prod_{\Gamma}s_{\Gamma}$ be a
primary decomposition, so that $V_{+}=\bigoplus_{\Gamma}V_{\Gamma}$ with
$V_{\Gamma}$ the underlying space of $s_{\Gamma}$. Thus

$C_{G_{+}}(s_{+})=\prod_{\Gamma}C_{\Gamma},\quad C_{\Gamma}={\rm
GL}^{\epsilon_{\Gamma}}(m_{\Gamma},q^{\delta_{\Gamma}})$ $None$

with $m_{\Gamma}=m_{\Gamma}(s_{+})$. We may suppose $D_{+}\in{\rm
Syl}_{p}(C_{G_{+}}(s_{+}))$, that is, a Sylow subgroup of $C_{G_{+}}(s_{+})$,
so that

$D_{+}=\prod_{\Gamma}D_{\Gamma},\quad D_{\Gamma}\in{\rm Syl}_{p}(C_{\Gamma}).$
$None$

So $D_{G}$ is a direct product of cyclic groups and wreath product $p$-groups.
Since $C_{V_{\Gamma}}(D_{\Gamma})=\\{0\\}$, it follows that
$m_{\Gamma}=e_{\Gamma}w_{\Gamma}$ for some integer $w_{\Gamma}$, and

$D_{\Gamma}\in{\rm Syl}_{p}({\rm
GL}^{\epsilon_{\Gamma}}(w_{\Gamma},q^{e_{\Gamma}\delta_{\Gamma}})).$

Suppose $e\neq 1$ or $\eta\neq\epsilon$, so that $Z(G)=C_{q-\eta}$ is a
$p^{\prime}$-group. Thus $D\cong D_{u}=D_{G}$. In particular, $D_{G}$ is
abelian and each $w_{\Gamma}<p$. Thus
$D_{\Gamma}\cong(C_{p^{a+\alpha_{\Gamma}}})^{w_{\Gamma}}$. Note that
$w_{\Gamma}\geq 1$ for some $\Gamma$. Since $D_{\Gamma}\leq C_{p}\times
C_{p}$, we have $a=1$ and $\alpha_{\Gamma}=0$.

Suppose $e=1$ and $\eta=\epsilon$, so that $(q-\epsilon)_{p}=p^{a}$ and each
$e_{\Gamma}=1$. If $w_{\Gamma}\geq 2$, then define
$R_{\Gamma}=\langle{\mathrm{diag}}(x,x^{-1},I_{w_{\Gamma}-2})\mid x\in
C_{p^{a+\alpha_{\Gamma}}}\rangle$, a subgroup of $D_{\Gamma}$, with
$R_{\Gamma}\leq S(V_{\Gamma})$ and $R_{\Gamma}\cap Z(S(V_{\Gamma}))=1$. So
$R_{\Gamma}\leq D_{u}$ and $R_{\Gamma}\cong R_{\Gamma}Z/Z\leq D$. Since
$R_{\Gamma}\cong C_{p^{a+\alpha_{\Gamma}}}$, we have that $a=1$ and
$\alpha_{\Gamma}=0$.

Hence it suffices to consider $\Gamma$ with $w_{\Gamma}=1$, $D_{\Gamma}\cong
C_{p^{a+\alpha_{\Gamma}}}$ and $\dim V_{\Gamma}=d_{\Gamma}$. In particular,
$D_{G}$ is abelian.

If $D_{G}$ is cyclic, then so is $D_{G}Z/Z$. Thus a defect group of any block
of $G/Z$ covering $B$ is cyclic. Suppose $D_{G}$ is non-cyclic.

Note that $C_{\Gamma}\cong{\rm GL}^{\epsilon_{\Gamma}}(1,q^{\delta_{\Gamma}})$
is a torus and

$C_{\Gamma}\cap S(V_{\Gamma})\cong
C_{q^{\delta_{\Gamma}-1}+\epsilon_{\Gamma}q^{\delta_{\Gamma}-2}+\ldots+1}.$

If $\alpha_{\Gamma}=0$, then $p\nmid\delta_{\Gamma}$,
$D_{\Gamma}=Z(G(V_{\Gamma}))_{p}=C_{p^{a}}$ and so $D_{\Gamma}\cap
S(V_{\Gamma})=1$.

Since $D_{G}$ is noncyclic, we have $m_{\Delta}=w_{\Delta}=1$ for some
$\Delta\neq\Gamma$. Now $\det:C_{\Gamma}\times
C_{\Delta}\to\mathbb{F}_{q}^{\times}$ is surjective, so $\det:D_{\Gamma}\times
D_{\Delta}\to C_{p^{a}}=O_{p}(\mathbb{F}_{q}^{\times})$ is onto. In
particular, $(D_{\Gamma}\times D_{\Delta})\cap S(V_{\Gamma}\perp V_{\Delta})$
contains a cyclic group $P=\\{{\mathrm{diag}}(x,y)\mid x\in
D_{\Gamma},y=y(x)\in D_{\Delta}\\}$, where $\det(y(x))=\det(x)^{-1}$. Since
$\det(x)\neq 1$, we have that $PZ/Z=P\leq D$. In particular, $a=1$.

Suppose each $\alpha_{\Gamma}\geq 1$. Then
$D_{\Gamma}/Z(G(V_{\Gamma}))_{p}\cong C_{p^{\alpha_{\Gamma}}}$. A proof
similar to above shows that $\alpha_{\Gamma}\leq 1$. Thus $\alpha_{\Gamma}=1$
and $(\delta_{\Gamma})_{p}=p$. So $D_{\Gamma}\cap S(V_{\Gamma})\cong C_{p}$.
Since $|(D_{\Gamma}\times D_{\Delta})\cap S(V_{\Gamma}\perp V_{\Delta})|\geq
p^{a+\alpha_{\Gamma}+\alpha_{\Delta}}$ and $D=D_{u}Z/Z=C_{p}\times C_{p}$, we
have $m_{\Gamma^{\prime}}(s_{+})=0$ for
$\Gamma^{\prime}\not\in\\{\Gamma,\Delta\\}$, so

$D_{+}=D_{\Gamma}\times D_{\Delta}\cong C_{p^{a+\alpha_{\Gamma}}}\times
C_{p^{a+\alpha_{\Delta}}}=C_{p^{a+1}}\times C_{p^{a+1}}.$

Let $Z_{\Gamma}=Z(G(V_{\Gamma}))_{p}\cong C_{p^{a}}$. Suppose $a\geq 2$. A
similar proof to above shows that $D_{+}\cap S(V_{+})$ contains a subgroup
$P=\\{{\mathrm{diag}}(x,y)\mid x\in Z_{\Gamma},y=y(x)\in D_{\Delta}\\}$, where
$\det(y(x))=\det(x)^{-1}$. Since $(d_{\Gamma})_{p}=p$, we have $\det(x)\neq 1$
for generators $x\in Z_{\Gamma}$, so $P\cap Z(S(V_{+}))=1$. Hence $PZ/Z\cong
P\leq D$, which is impossible. So $a=1$. $\Box$

Let $V$ be a non-degenerate orthogonal or symplectic space, $G=I_{0}(V)$ and
let $G^{*}$ be the dual group of $G$. Note that

${\rm Sp}_{2n}(q)^{*}={\rm SO}_{2n+1}(q),\quad{\rm SO}_{2n+1}(q)^{*}={\rm
Sp}_{2n}(q),\quad{\rm SO}_{2n}^{\eta}(q)^{*}={\rm SO}_{2n}^{\eta}(q).$

If $B$ is a $p$-block of $I_{0}(V)$, then there exists a semisimple
$p^{\prime}$-element $s\in I_{0}(V)^{*}$ such that

$B\subseteq\mathcal{E}_{p}(I_{0}(V),(s)).$

Let $(D,b_{D})$ be a Sylow $B$-subgroup of $I_{0}(V)$. Then $V$ and $D$ have
corresponding decompositions

$V=V_{0}\perp V_{+},\quad D=D_{0}\times D_{+}.$ $None$

If $p$ is odd, then $V_{0}=C_{V}(D)$, $V_{+}=[V,D]$, $D_{0}=\\{1_{V_{0}}\\}$
and $D_{+}\leq I_{0}(V_{+})$. Let $G_{0}=I_{0}(V_{0})$, $G_{+}=I_{0}(V_{+})$,
$C_{+}=C_{I_{0}(V_{+})}(D_{+})$ and let $V^{*}$ be the underlying space of
$I_{0}(V)^{*}$.

Let $z\in D$ be a primitive element. Then

$z=z_{0}\times z_{+},\quad L=C_{G}(z)=L_{0}\times L_{+},\quad
L_{0}=G_{0},\quad L_{+}={\rm GL}^{\epsilon}(m,q^{e}),$ $None$

where $z_{0}=1_{V_{0}}$, $z_{+}\leq D_{+}$ and $\dim V_{+}=2em$. Then $L$ is a
Levi subgroup of $G$ and we may suppose $s\in L^{*}\leq G^{*}$. In particular,

$V^{*}=U_{0}\perp U_{+}\quad\hbox{ and }\quad s=s_{0}\times s_{+},$ $None$

where $U_{0}=V_{0}^{*}$, $s_{0}\in L_{0}^{*}=I_{0}(U_{0})$, $U_{+}$ is the
underlying space of $L_{+}^{*}$ and $s_{+}\in L_{+}^{*}\leq I_{0}(U_{+})$.

###### Proposition 4.2

Let $p$ be an odd prime

(a) Suppose $K_{c}\in\\{{\rm
Sp}_{2n}(q),\Omega^{\eta}_{2n}(q),\Omega_{2n+1}(q)\\}$ and $K=K_{c}/Z_{c}$ for
some $Z_{c}\leq Z(K_{c})$. If $B\in{\rm Blk}(K)$ with $D_{K}=D(B)\cong C_{p}$
or $C_{p}\times C_{p}$, then $a=1$. In particular, $K$ has no field
automorphism of order $p$.

(b) let $V$ be orthogonal, $K_{u}={\rm Spin}^{\eta}(V)$ and $K=K_{u}/Z_{u}$
for some $Z_{u}\leq Z(K_{u})$, where $\eta=\circ$ when $\dim V$ is odd.
Suppose $B_{K}\in{\rm Blk}(K)$ with $D_{K}=D(B_{K})\cong C_{p}$ or
$C_{p}\times C_{p}$. Then $a=1$ and $K$ has no field automorphism of order
$p$.

(c) Let $V$ be orthogonal such that $K_{u}={\rm Spin}_{8}^{+}(q)={\rm
Spin}(V)$, $Z_{u}=1$ or $Z(K_{u})$ and $K=K_{u}/Z_{u}$. Let $\Phi_{K}$ and
$\Gamma_{K}$ be the field and graph automorphism groups of $K$, respectively,
so that $\Phi_{K}\cong C_{f}$ and $\Gamma_{K}\cong S_{3}$. Let

$K_{u}\leq A\leq K_{u}.{\rm Out}(K),$

and $A_{u}=A/Z_{u}$, where ${\rm Out}(K)=\Phi_{K}\times{\rm
Outdiag}(K_{u}).\Gamma_{K}$ with ${\rm Outdiag}(K).\Gamma_{K}\cong S_{3}$ or
$S_{4}$ according to $r=2$ or odd.

Suppose $p=3$ and $D\leq A_{u}$ is a defect group such that $D\cong
3_{+}^{1+2}$ and $D_{K}=D\cap K\cong C_{3}$ or $C_{3}\times C_{3}$.

(i) Then $C_{K}(D)={\rm SL}_{3}^{\epsilon}(q)$ and

$N_{A_{u}}(D)=(3_{+}^{1+2}\circ_{3}{\rm SL}_{3}^{\epsilon}(q)).(C_{d}\times
S_{3}\times\Phi)$

where $C_{d}=1$ or $C_{2}$ and $\Phi$ is a $3^{\prime}$-subgroup of
$\Phi_{K}$. Moreover, the outer side group $S_{3}$ in $N_{A_{u}}(D)$ is
generated by $x$ and $y$ with $|x|=3$ and $|y|=2$ such that $x$ and $y$ induce
outer-diagonal and graph automorphisms on ${\rm SL}_{3}^{\epsilon}(q)$,
respectively.

(ii) The subgroup $D\cong 3_{+}^{1+2}$ given above is not a defect group of
$A_{u}$.

Proof: (a) Let $V$ the underlying space of $K_{c}$ and $B_{c}\in{\rm
Blk}(K_{c})$ dominating $B$. Then $D_{c}\cong D_{c}Z_{c}/Z_{c}\cong D_{K}\cong
C_{p}$ or $C_{p}\times C_{p}$ for some $D_{c}=D(B_{c})$. Let $G=I_{0}(V)$ and
$B_{G}\in{\rm Blk}(G)$ covering $B_{c}$. Then $D_{c}=D\cap K_{c}$ for some
$D=D(B_{G})$. Since $I_{0}(V)/K_{c}$ is a $p^{\prime}$-group, we have
$D_{c}=D$.

Follow the notation above. We have $D=D_{0}\times D_{+}\cong D_{+}$. Identify
$L_{+}$ with $L_{+}^{*}$. Then $D_{+}\in{\rm Syl}_{p}(C_{L_{+}}(s_{+}))$. In
particular, $C_{p^{a}}\cong Z(L_{+})_{p}\leq D_{+}$ and hence $a=1$.

(b) Let $Z\leq Z(K_{u})$ such that $K_{c}=\Omega^{\eta}(V)=K_{u}/Z$ and let
$B_{u}\in{\rm Blk}(K_{u})$ dominate $B$ and $B_{c}\in{\rm Blk}(K_{c})$
dominated by $B_{u}$, Then $D_{K}=D_{u}Z_{u}/Z_{u}$ for some $D_{u}=D(B_{u})$
and $D_{u}Z/Z=D(B_{c})$. Since $Z$ and $Z_{u}$ are $p^{\prime}$-group, we have
$D(B_{c})\cong D_{u}\cong D_{K}\cong C_{p}$ or $C_{p}\times C_{p}$. By
Proposition 4.2 (a), $a=1$.

(c) (i) Let $B$ be a block of $A_{u}$ with defect group $D$ and let $B_{K}$ be
a block of $K$ covered by $B$. We may suppose $D(B_{K})=D_{K}$, wich is
isomorphic to $C_{3}$ or $C_{3}\times C_{3}$. By parts (a) and (b) above, $K$
has no field automorphism of order $p$. So we may suppose $A\leq K_{u}.S_{3}$
or $A\leq K_{u}.S_{4}$ according to $r=2$ or odd.

Since a Sylow $3$-subgroup of ${\rm Out}(K)$ has order $3$, we have that
$D_{K}\cong C_{3}\times C_{3}$ and $Z(D)\leq D_{K}$. Thus $D$ contains an
order $3$ graph automorphism $\gamma_{K}$ of $K$. Note that $\gamma_{K}$ acts
on $Z(K_{u})\setminus\\{1\\}$ transitively. So we may assume that
$Z_{u}=Z(K_{u})$ and $K={\rm P}\Omega_{8}^{+}(q)$. By [27, Table 4.7.3A],

$C_{K}(Z(D))=(C_{q-\epsilon}\times C_{q-\epsilon})\circ_{3}({\rm
SL}_{3}^{\epsilon}(q).x),$

where $x$ induces the order $3$ outer diagonal automorphism on ${\rm
SL}_{3}^{\epsilon}(q)$. In addition, $N_{K}(Z(D))=C_{K}(Z(D)).y$ and

${\rm Out}_{{\rm
Outdiag}(K).S_{3}}(Z(D))\cong\langle\gamma_{K},\rho_{K}\rangle\cong S_{3},$

where $y$ induces a graph automorphism on ${\rm SL}_{3}^{\epsilon}(q)$ and
inveres elements of $(C_{q-\epsilon}\times C_{q-\epsilon})$, $\gamma_{K}$
induces $(\omega{:}1)$ on $C_{K}(Z(D))$ (there is a typo in [27, Table
4.7.3A], see [27, Table 4.7.2]) and
$\langle\gamma_{K},\rho_{K}\rangle=\Gamma_{K}$. So
$D_{K}=O_{3}(C_{q-\epsilon}\times C_{q-\epsilon})$ and $D=\langle
D_{K},\gamma_{K}\rangle\cong 3_{+}^{1+2}$. Since $N_{A_{u}}(D))\leq
N_{A_{u}}(Z(D))$, it follows that $C_{A_{u}}(D)={\rm SL}_{3}^{\epsilon}(q)$
and

$N_{A_{u}}(D)=(3_{+}^{1+2}\circ_{3}{\rm SL}_{3}^{\epsilon}(q)).(\langle
x,y\rangle\times\langle\rho\rangle),$

where $\rho=1$ or $\rho_{K}$ according to $\rho_{K}\not\in A_{u}$ or
$\rho_{K}\in A_{u}$. Thus (c) (i) holds.

(ii) Follow the notation of (i). Let $(D,b_{D})$ be a Sylow $B$-subgroup, so
that $b_{D}$ is a block of $C_{A_{u}}(D)={\rm SL}_{3}^{\epsilon}(q)$. If
$\theta$ is the canonocal character of $b_{D}$, then $Z(C_{A_{u}}(D))$ is in
the kernel of $\theta$ and $\theta$ has $3$-defect $0$ as character of
$C_{A_{u}}(D)/Z(C_{A_{u}}(D))$. By [6, Lemma 6.12], $\theta$ extends to ${\rm
PGL}_{3}^{\epsilon}(q)$. Since $x\in N_{A_{u}}(D)$ induces an order $3$
diagonal outer automorphism on ${\rm SL}_{3}^{\epsilon}(q)$, it follows that
$x\in N_{A_{u}}(D,b_{D})$. In particular, $N_{A_{u}}(D,b_{D})/(C_{A_{u}}(D)D)$
is not a $3^{\prime}$-group and hence $(D,b_{D})$ is not a Sylow $B$-subgroup,
which is impossible. This proves (c) (ii). $\Box$

##  5 Exceptional groups

We repeat the analysis in the previous section for the exceptional groups. In
this case we cannot rule out the existence of a field automorphism of order
$p$, but the cases in which one may occur are limited, and must centralise the
defect groups.

We will follow the notation of [27]. Let $K$ be a quasisimple group of
exceptional type and $B\in{\rm Blk}(K)$ with defect group $D\cong C_{p}\times
C_{p}$. Let $({\overline{K}},\sigma)$ be a $\sigma$-setup for $K$ in the sense
of [27, Definition 2.2.1], and let ${\overline{T}}\leq{\overline{K}}$ be a
maximal torus. A subgroup $A\leq K$ is toral if $A$ is a subgroup of a maximal
torus of ${\overline{K}}$. Two elements $w,y\in{\overline{K}}$ are
$\sigma$-conjugate if $w=xy\sigma(x)^{-1}$ for some $x\in{\overline{K}}$.

We need the following proposition

###### Proposition 5.1

([3, Propositions 4.1, 4.3 and 4.4 and Lemma 5.2])

Let $A\leq K$ be a subgroup. Then the following hold.

  1. $(a)$

There is a bijection between $({\overline{K}})^{\sigma}$-classes of subgroups
of $({\overline{K}})^{\sigma}$ which are ${\overline{K}}$-conjugate to $A$,
and $\sigma$-classes in $N_{{\overline{K}}}(A)/C_{{\overline{K}}}(A)^{\circ}$
contained in $C_{{\overline{K}}}(A)/C_{{\overline{K}}}(A)^{\circ}$: the
$\sigma$-class of $w\in C_{{\overline{K}}}(A)/C_{\overline{K}}(A)^{\circ}$
corresponds to the $({\overline{K}})^{\sigma}$-class of subgroups with
representative $A_{w}={{}^{g}}A$, where $g\in{\overline{K}}$ is chosen with
$g^{-1}\sigma(g)C_{{\overline{K}}}(A)^{\circ}=w$.

  2. $(b)$

Let $A_{w}\leq({\overline{K}})^{\sigma}$ be the ${\overline{K}}$-conjugate of
$A$ as in $(a)$. If $\dot{w}\in C_{\overline{K}}(A)$ is any lift of $w$, then

$(C_{{\overline{K}}}(A_{w})^{\circ})^{\sigma}\cong(C_{{\overline{K}}}(A)^{\circ})^{\dot{w}\sigma}$

where $(C_{{\overline{K}}}(A)^{\circ})^{\dot{w}\sigma}$ is the set of fixed
points of $\dot{w}\sigma$ in $C_{{\overline{K}}}(A)^{\circ}$.

  3. $(c)$

If $A_{w}$ is as in $(a)$ and $w\sigma$ is identified with the map $x\mapsto
w\sigma(x)w^{-1}$, then

$(N_{{\overline{K}}}(A_{w}))^{\sigma}/(C_{{\overline{K}}}(A_{w})^{\circ})^{\sigma}\cong(N_{{\overline{K}}}(A)/C_{{\overline{K}}}(A)^{\circ})^{w\sigma}$

and
$C_{\overline{G}}(A_{w})^{\sigma}/(C_{{\overline{K}}}(A_{w})^{\circ})^{\sigma}\cong(C_{{\overline{K}}}(A)/C_{{\overline{K}}}(A)^{\circ})^{w\sigma}$

  4. $(d)$

If, moreover that $A\leq{\overline{T}}\cap K$ is toral, then
$N_{\overline{K}}(A)/C_{\overline{K}}(A)^{\circ}=N_{{\overline{K}}^{\sigma}}(A)/(C_{\overline{K}}(A)^{\circ})^{\sigma}$,
$C_{\overline{K}}(A)/C_{\overline{K}}(A)^{\circ}=C_{{\overline{K}}^{\sigma}}(A)/(C_{\overline{K}}(A)^{\circ})^{\sigma}$
and $\sigma$ acts trivially on
$N_{\overline{K}}(A)/C_{\overline{K}}(A)^{\circ}$.

Thus if $\sigma$ acts as identity on
$H=N_{{\overline{K}}}(A)/C_{{\overline{K}}}(A)^{\circ}$, then
$H^{w\sigma}=C_{H}(w)$ and the $\sigma$-classes are the $H$-conjugacy classes.
So toral elementary abelian $p$-subgroups of a finite exceptional group of Lie
type can be determined by [3] and [4] (cf. [3, Section 5]). If $D=\langle
x,y\rangle\cong C_{p}\times C_{p}$ and $K=K_{u}$, then $C_{\overline{K}}(x)$
is connected and so $y$ lies in a maximal torus of $C_{\overline{K}}(x)$. In
particular, $D$ is toral.

###### Proposition 5.2

Let $p$ be an odd prime and let $K$ be a finite exceptional quasisimple group
and $B\in{\rm Blk}(K)$ with defect group $D\cong C_{p}\times C_{p}$ and Sylow
$B$-subgroup $(D,b_{D})$. Then one of the following holds.

$(a)$ $a=1$.

$(b)$ $p=5$, $K=E_{8}(q)$, $e=1$ and $B=\mathcal{E}_{5}(K,(s))$, where
$s\in\mathcal{F}^{5^{\prime}}$ such that $C_{K^{*}}(s)=C_{q^{4}+\epsilon
q^{3}+q^{2}+\epsilon q+1}\times C_{q^{4}+\epsilon q^{3}+q^{2}+\epsilon q+1}$.

$(c)$ $p=3$,
$K\in\\{{}^{3}D_{4}(q),F_{4}(q),(E_{6})_{u}^{-\epsilon}(q),(E_{7})_{u}(q)/Z,E_{8}(q)\\}$
with $Z\leq Z((E_{7})_{u}(q))$ such that

$C_{K}(D)^{\circ}=\left\\{\begin{array}[]{lll}C_{q^{2}+\epsilon q+1}\times
C_{q^{2}+\epsilon q+1}&{\rm{if}}&K\in\\{{}^{3}D_{4}(q),F_{4}(q)\\},\\\
C_{q^{2}+\epsilon q+1}\times
C_{q^{4}+q^{2}+1}&{\rm{if}}&K=(E_{6})_{u}^{-\epsilon}(q),\\\ C_{q^{2}+\epsilon
q+1}\times C_{q^{2}+\epsilon q+1}\times({\rm
SL}_{2}(q^{3})/Z)&{\rm{if}}&K=(E_{7})_{u}(q)/Z,\\\ C_{q^{2}+\epsilon
q+1}\times C_{q^{2}+\epsilon
q+1}\times{}^{3}D_{4}(q)&{\rm{if}}&K=E_{8}(q),\end{array}\right.$

and $b_{D}$ covers a block $b\in{\rm Blk}(C_{K}(D)^{\circ})$ with $D(b)=D$,
where
$C_{K}(D)^{\circ}=(C_{\overline{K}}(D)^{\circ})^{\sigma}=C_{C_{\overline{K}}(D)^{\circ}}(\sigma)$
and $C_{K}(D)=C_{K}(D)^{\circ}.p$ except when $K={}^{3}D_{4}(q)$, in which
case $C_{K}(D)=C_{K}(D)^{\circ}$.

Proof: Let $K_{u}$ be a universal group such that $K=K_{u}/Z$ for some $Z\leq
Z(K_{u})$, and let $B_{u}\in{\rm Blk}(K_{u})$ dominate $B$ and
$D_{u}=D(B_{u})$ such that $D=D_{u}Z/Z$. If
$Z(K_{u})\neq\Omega_{1}(Z(D_{u}))$, then take $z\in Z(D_{u})\setminus
Z(K_{u})$ with $|z|=p$. If $Z(K_{u})=\Omega_{1}(Z(D_{u}))$, then take $z\in
D_{u}$ such that $zZ(K_{u})\in Z(D_{u}/Z(K_{u}))$. Let $(z,B_{z})$ be a
$B_{u}$-subsection, and suppose it is major when $z\in Z(D_{u})$. In any case,
let $D_{z}$ be a defect group for $B_{z}$ with $D_{z}\leq D_{u}$. Then $B_{z}$
is a block of $C=C_{K_{u}}(z)$, and $D_{u}$ is a defect group for $B_{z}$ when
$(z,B_{z})$ is major. By [27, Theorem 4.2.2],

$C=O^{r^{\prime}}(C)T,\quad O^{r^{\prime}}(C)=L_{1}\circ L_{2}\circ\cdots\circ
L_{\ell}$

where each $L_{i}\in{\mathcal{L}ie}(r)$, and $T$ is an abelian
$r^{\prime}$-group inducing inner-diagonal automorphisms on each $L_{i}$. In
general, $z\not\in O^{r^{\prime}}(C)$, so we perform the following
modifications. If $Z(C)\leq O^{r^{\prime}}(C)$, then let $\ell=k$ and
$L=O^{r^{\prime}}(C)$. If $Z(C)\not\leq O^{r^{\prime}}(C)$, then let
$k=\ell+1$ and $L_{k}=Z(C)$. Thus

$C=LT\quad{\rm with}\quad\quad L:=L_{1}\circ L_{2}\circ\cdots\circ L_{k},$
$None$

$z\in Z(C)\leq L$ and $L\lhd C$. Let $B_{L}$ be a block of $L$ covered by
$B_{z}$. Note that $B_{L}$ has defect group $D_{L}=D_{z}\cap L$. If
$\chi\in{\rm Irr}(B_{L})$, then $\chi=\chi_{1}\circ\cdots\circ\chi_{k}$ for
some $\chi_{i}\in{\rm Irr}(L_{i})$, so that $\chi_{i}\in{\rm Irr}(B_{i})$ for
some $B_{i}\in{\rm Blk}(L_{i})$ and we write $B_{L}=B_{1}\circ
B_{2}\circ\cdots\circ B_{k}$.

Since $L=(L_{1}\times L_{2}\times\cdots\times L_{k})/A$ for a central subgroup
$A\leq(L_{1}\times L_{2}\times\cdots\times L_{k})$, it follows by [34,
Theorems 5.8.8 and 5.8.10] that

$D_{L}=(D(B_{1})\times D(B_{2})\times\cdots\times D(B_{k}))A/A$

where $D(B_{i})$ is some defect group of $B_{i}$ and $D(B_{i})$ is isomorphic
to a subgroup of $D_{L}$.

Since $z\in K_{u}$, it follows by [27, Theorem 4.1.9] that $z$ is of parabolic
or exceptional type. Suppose $z$ is of parabolic type. Then
${\overline{C}}=C_{{\overline{K}}_{u}}(z)=S{\overline{L}}$, where
${\overline{K}_{u}}$ is the corresponding algebraic group,
${\overline{L}}=[{\overline{C}},{\overline{C}}]$ is semisimple and
$S=Z({\overline{C}})$ is a torue. Since $z$ is parabolic, we have that $\dim
S\geq 1$ and hence $C_{q-\epsilon}\leq C_{S}(\sigma)\leq Z(C)$ except when
$p=3$ and $K=K_{u}={}^{3}D_{4}(q)$ and $C=(C_{q^{2}+\epsilon
q+1}\circ_{3}({\rm SL}_{3}^{\epsilon}(q).3))$, in which case
$Z(C)=C_{q^{2}+\epsilon q+1}$ (see [27, Table 4.7.3A]).

Suppose $(K_{u},p)\neq((E_{6})_{u}^{\epsilon}(q),3)$ and $C_{q-\epsilon}\leq
Z(C)$. Since $Z(C)$ is abelian and a defect group of $L$ is radical, we have
$C_{p^{a}}\leq D_{L}\leq D_{u}$. But $C_{p^{a}}\cap Z=1$, so
$C_{p^{a}}Z/Z=C_{p^{a}}\leq D$. Hence $a=1$.

Suppose $z$ is exceptional, so that $p$ is not good for $K$ in the sense of
[12, Table 1]. Thus $p\in\\{3,5\\}$ and moreover, $K=K_{u}=E_{8}(q)$ when
$p=5$. In particular, (a) holds when $p\geq 7$.

Suppose $D\cong D_{u}$ and $(K_{u},p)\neq((E_{6})_{u}^{\epsilon}(q),3)$. If
some $y\in D_{u}\setminus\\{1\\}$ is not exceptional, then replace $z$ by $y$
and repeat the proof above. We may suppose each $y\in D_{u}\setminus\\{1\\}$
is exceptional, except when $K=K_{u}={}^{3}D_{4}(q)$, in which case each $x\in
D_{u}\setminus\\{1\\}$ satisfies $C_{K}(x)\cong(C_{q^{2}+\epsilon
q+1}\circ_{3}({\rm SL}_{3}^{\epsilon}(q).3))$. Denote by
$T_{2,\epsilon}=C_{(q^{ep}-\epsilon)/(q^{e}-\epsilon)}\times
C_{(q^{ep}-\epsilon)/(q^{e}-\epsilon)}$.

(b) If $p=5$, then $K=K_{u}=E_{8}(q)$ and by [27, Table 4.7.3B],

$(e,C)\in\\{(1,({\rm SL}_{5}^{\epsilon}(q)\circ_{5}{\rm
SL}_{5}^{\epsilon}(q)).x),(2,{\rm SU}_{5}(q^{2}).5)\\},$

where $x=5{:}5$. Since $D_{u}\leq C$, we have $e=1$ and $C=({\rm
SL}_{5}^{\epsilon}(q)\circ_{5}{\rm SL}_{5}^{\epsilon}(q)).x$. In addition,
$D=\langle z,y\rangle$ for some $y\in C\setminus({\rm
SL}_{5}^{\epsilon}(q)\circ_{5}{\rm SL}_{5}^{\epsilon}(q))$ as $y$ is
exceptional. So

$C_{K}(D)=C_{K}(D)^{\circ}.5=T_{2,\epsilon}.5=(C_{q^{4}+\epsilon
q^{3}+q^{2}+\epsilon q+1})^{2}.5.$

By [27, Table 4.7.1], $\overline{C}={\rm SL}_{5}(\mathbb{F})\circ_{5}{\rm
SL}_{5}(\mathbb{F})$ is a connected reductive group, and by [23],
$B_{z}=\mathcal{E}_{5}(C,(t))$ for some $t\in\overline{C}^{*}$, where
$\mathbb{F}$ is the algebraic closure of $\mathbb{F}_{q}$. In addition,
$D=D(B_{z})$ is isomorphic to a Sylow subgroup of $C_{C^{*}}(t)$. Hence
$C_{\overline{C}^{*}}(t)={\overline{T}}^{*}$ is a maximal torus such that
${\overline{T}}^{\sigma}\cong T_{2,\epsilon}$, where ${\overline{T}}$ is the
dual of ${\overline{T}}^{*}$.

Take ${\overline{H}}_{1}={\rm
GL}_{5}(\mathbb{F})={\overline{T}}_{1}\circ_{5}{\rm SL}_{5}(\mathbb{F})$ and
${\overline{H}}_{2}={\rm SL}_{5}(\mathbb{F})\circ_{5}{\overline{T}}_{1}$ be
subgroups of $\overline{C}$. Then each ${\overline{H}}_{i}$ is a Levi subgroup
of both $\overline{C}$ and ${\overline{K}}$, and
$C_{\overline{C}^{*}}(t)\leq{\overline{H}}_{i}^{*}$. If
$H_{i}={\overline{H}}_{i}^{\sigma}$, then we may suppose $t\in H_{i}^{*}\leq
K^{*}$. Since each $C_{H_{i}^{*}}(t)=T_{2,\epsilon}^{*}$, it follows that
$C_{K^{*}}(t)=T_{2,\epsilon}^{*}$. If $C_{K^{*}}(t)=T_{2,\epsilon}^{*}$, then
$b=\mathcal{E}_{p}(T_{2,\epsilon},(t))$ is a block with a defect group
$O_{p}(T_{2,\epsilon})=C_{p}\times C_{p}$. But $T_{2,\epsilon}$ is a Levi
subgroup and $R_{T_{2,\epsilon}}^{K}$ is a perfect isometry, so
$B=\mathcal{E}_{p}(K,(t))$ is a block and $D(B)\cong C_{p}\times C_{p}$. Here
$R_{T_{2,\epsilon}}^{K}$ is the Deligne-Lusztig map from
$\mathcal{E}_{p}(T_{2,\epsilon},(t))$ to $\mathcal{E}_{p}(K,(t))$. Similarly,
$R_{T_{2,\epsilon}}^{C}(b)=B_{z}$. Thus (b) holds.

(c) Suppose $p=3$, so that $e=1$. The proof is similar to that of (b). Let
$K=K_{u}={}^{3}D_{4}(q)$, so that
$\overline{C}={\overline{T}}_{2}\circ_{3}{\rm SL}_{3}(\mathbb{F})$ is Levi.
Thus $C=C_{q^{2}+\epsilon q+1}\circ_{3}({\rm SL}_{3}^{\epsilon}(q).3)$ and
$B_{z}=\mathcal{E}_{p}(C,(s))$ for some $s\in(\overline{C}^{*})^{\sigma}$. So
$D=D_{u}=C_{p}\times C_{p}$ and $D=\langle z,y\rangle$ for some $y\in({\rm
SL}_{3}^{\epsilon}(q).3)\setminus{\rm SL}_{3}^{\epsilon}(q)$. Thus
$C_{K}(D)=C_{C}(y)=T_{2,\epsilon}=(C_{q^{2}+\epsilon q+1})^{2}$ and $s\in
T_{2,\epsilon}^{*}$. A proof similar to that of (b) shows that
$C_{K^{*}}(s)=T_{2,\epsilon}^{*}$ (cf. [2, Table 1]). Conversely, if $s\in
K^{*}$ such that $C_{K^{*}}(s)=T_{2,\epsilon}^{*}$, then
$B=\mathcal{E}_{p}(K,(s))$ is a block and $D(B)=O_{p}(T_{2,\epsilon})\cong
C_{p}\times C_{p}$. Thus (c) holds.

Suppose $K=F_{4}(q)$, so that $K=K_{u}$, $\overline{C}={\rm
SL}_{3}(\mathbb{F})\circ_{3}{\rm SL}_{3}(\mathbb{F})$, $C=({\rm
SL}_{3}^{\epsilon}(q)\circ_{3}{\rm SL}_{3}^{\epsilon}(q)).x$ and
$D=D_{u}=C_{p}\times C_{p}$, where $x=3{:}3$. A proof similar to that of (b)
shows that $C_{K}(D)=T_{2,\epsilon}.3=(C_{q^{2}+\epsilon q+1})^{2}.3$,
$B_{z}=\mathcal{E}_{p}(C,(s))$ and $s\in T_{2,\epsilon}^{*}$. In addition,
$C_{K^{*}}(s)=T_{2,\epsilon}^{*}$ and $B=\mathcal{E}_{p}(K,(s))$ is a block
with defect group $D=O_{p}(T_{2,\epsilon})=C_{p}\times C_{p}$.

If $K=K_{u}=(E_{6})_{u}^{-\epsilon}(q)$ and $z$ is exceptional, then
$\overline{C}=({\rm SL}_{3}(\mathbb{F}))^{3}/3$ and by [27, Table 4.7.3A],
$C=({\rm SL}_{3}^{\epsilon}(q)\circ_{3}{\rm SL}_{3}(q^{2})).x$ with $x=3{:}3$.
Thus $D=\langle z,y\rangle$ for some $y\in C\setminus{\rm
SL}_{3}^{\epsilon}(q)\circ_{3}{\rm SL}_{3}(q^{2})$, and
$C_{K}(D)=C_{K}(D)^{\circ}.3=(C_{q^{2}+\epsilon q+1}\times
C_{q^{4}+q^{2}+1}).3$. A proof similar to that of (b) shows that
$B_{z}=\mathcal{E}_{p}(C,(s))$, $s\in(C_{q^{2}+\epsilon q+1}\times
C_{q^{4}+q^{2}+1})^{*}$ and $C_{K^{*}}(s)\cong C_{q^{2}+\epsilon q+1}\times
C_{q^{4}+q^{2}+1}$. Conversely, if $C_{K^{*}}(s)\cong C_{q^{2}+\epsilon
q+1}\times C_{q^{4}+q^{2}+1}$, then $B=\mathcal{E}_{p}(K,(s))$ is a block with
defect group $D=O_{p}(C_{q^{2}+\epsilon q+1}\times
C_{q^{4}+q^{2}+1})=C_{p}\times C_{p}$.

If $K_{u}=(E_{7})_{u}(q)$, then by [27, Table 4.7.3A], $C=({\rm
SL}_{3}^{\epsilon}(q)\circ_{6^{*}}{\rm SL}_{6}^{\epsilon}(q).6^{*}).x$, where
$6^{*}=\gcd(6,q-\epsilon)$ and $x=3{:}3$. Note that $D=D_{u}Z/Z\cong D_{u}$.
So $D=\langle z,y\rangle$ for some $y\in C\setminus({\rm
SL}_{3}^{\epsilon}(q)\circ_{6^{*}}{\rm SL}_{6}^{\epsilon}(q).6^{*})$. Hence
$C_{K_{u}}(D)=C_{K_{u}}(D)^{\circ}.p=(T_{2,\epsilon}\times{\rm
SL}_{2}(q^{3})).p$. If $(D,b_{D})$ is a Sylow $B$-subgroup, then $b_{D}$
covers a block $b=b_{1}\times b_{2}\in{\rm Blk}(C_{K_{u}}(D)^{\circ})$, where
$b_{1}\in{\rm Blk}(T_{2,\epsilon})$ with $D(b_{1})=D$ and $b_{2}\in{\rm
Blk}({\rm SL}_{2}(q^{3}))$ with $D(b_{2})=1$. Conversely, suppose
$b=b_{1}\times b_{2}\in{\rm Blk}(C_{K_{u}}(D)^{\circ})$ such that
$b_{1}\in{\rm Blk}(T_{2,\epsilon})$ with $D(b_{1})=O_{p}(T_{2,\epsilon})=D$
and $b_{2}\in{\rm Blk}({\rm SL}_{2}(q^{3}))$ with $D(b_{2})=1$. Note that
$C_{K_{u}}(D)^{\circ}$ is a Levi subgroup of $K_{u}$, so
$B=R_{C_{K_{u}}(D)^{\circ}}^{K_{u}}(b)$ is a block of $K_{u}$ with $D(B)=D$.
Thus (c) holds. If $Z=Z(K_{u})\neq 1$, then $Z=C_{2}$,
$C_{K}(D)=C_{K_{u}}(D_{u})/Z=(T_{2,\epsilon}\times{\rm SL}_{2}(q^{3})/Z).p$. A
proof similar to that of $K_{u}=(E_{7})_{u}(q)$ shows that (c) also holds.

If $K=K_{u}=E_{8}(q)$, then by [27, Table 4.7.3A],

$C\in\\{({\rm SL}_{9}^{\epsilon}(q)/3).(9^{*}/3),({\rm
SL}_{3}^{\epsilon}(q)\circ_{3}(E_{6})_{u}(q)).x\\}$

with $x=3{:}3$. By [27, Table 4.7.3A] again,
$C_{K}(D)=C_{K}(D)^{\circ}.p=(T_{2,\epsilon}\times{}^{3}D_{4}(q)).p$. A proof
similar to that of $K_{u}=(E_{7})_{u}(q)$ shows that the root block
$b_{D}\in{\rm Blk}(C_{K}(D))$ of $B$ covers a block $b=b_{1}\times
b_{2}\in{\rm Blk}(C_{K}(D)^{\circ})$, where $b_{1}\in{\rm
Blk}(T_{2,\epsilon})$ with $D(b_{1})=D$ and $b_{2}\in{\rm
Blk}({}^{3}D_{4}(q))$ with $D(b_{2})=1$. Conversely, if $b\in{\rm
Blk}(C_{K}(D)^{\circ})$ with $D(b)=D$, then $B=R_{C_{K}(D)^{\circ}}^{K}(b)$
has a defect group $D$. So (c) holds.

If $K=K_{u}=(E_{6})_{u}^{\epsilon}(q)$ and $z$ is parabolic, then by [27,
Table 4.7.3A], $C_{q-\epsilon}\leq Z(C)$ and $C_{p^{a}}\leq D$ as $D$ is a
defect group of $C$. Hence $a=1$ and (a) holds. If $z$ is exceptional, then
$C=(({\rm SL}_{3}^{\epsilon}(q)\times{\rm SL}_{3}^{\epsilon}(q)\times{\rm
SL}_{3}^{\epsilon}(q))/3).x$ such that $x=3{:}3{:}3$. Since $D\leq C$ is a
defect group, we have $Z(C)=C_{p}\times C_{p}\leq D$ and hence $D=Z(C)$. Thus
$C_{K_{u}}(D)=C$. So $L=({\rm SL}_{3}^{\epsilon}(q)\times{\rm
SL}_{3}^{\epsilon}(q)\times{\rm SL}_{3}^{\epsilon}(q))/3$ and
$B_{L}=B_{1}\circ B_{2}\circ B_{3}$ with $D(B_{L})=D(B_{1})\circ D(B_{2})\circ
D(B_{3})=D$. Each $B_{i}\in{\rm Blk}({\rm SL}_{3}^{\epsilon}(q))$ with
$D(B_{i})=C_{3}=Z({\rm SL}_{3}^{\epsilon}(q))$. Since $x=3{:}3{:}3$ induces
inner-diagonal automorphism on each factor ${\rm SL}_{3}^{\epsilon}(q)$, we
have that $x$ stabilises each $B_{i}$ and hence $x$ stabilises $B_{L}$. If
$(D,b_{D})$ is a Sylow $B$-subgroup such that $b_{D}$ covers $B_{L}$, then
$D\neq D(b_{D})$, which is impossible.

If $D\not\cong D_{u}$, then $(K,p)=((E_{6})_{a}^{\epsilon}(q),3)$ and
$Z(K_{u})=C_{3}\leq D_{u}$. In this case, we take $z\in D$ and let $(z,B_{z})$
be a major subsection of $B$. Hence $D=D(B_{z})$ and $D\leq C_{K}(z)$ is a
defect group. If $z$ is of parabolic type, then by [27, Table 4.7.3A],
$C_{q-\epsilon}\leq Z(C_{K}(z))$ and hence $C_{p^{a}}\leq D$ as $D$ is radical
of $C_{K}(z)$. In particular, $a=1$. Note that if $C_{K}(u)\cong
C_{q^{2}+\epsilon q+1}\times{}^{3}D_{4}(q)$ for some order $3$ element $u\in
K$ given in [27, Table 4.7.3A], then $u$ is diagonal, which is impossible as
$K$ is quasisimple. If $z$ is of exceptional type, then $C_{K}(z)=({\rm
SL}_{3}^{\epsilon}(q)\times{\rm SL}_{3}^{\epsilon}(q)\times{\rm
SL}_{3}^{\epsilon}(q))/(3^{2}).x.t$, where $x=3{:}3{:}1$ and $t=1{:}3{:}3$.
Thus $D\leq C_{K}(z)$ contains always an order $3$ element of parabolic type.
Hence (a) holds. $\Box$

###### Remark 5.3

$(a)$ Let $p=5$ and $K_{0}=E_{8}(r)\leq K=E_{8}(q)$ such that $r\equiv 1$ mod
$5$ and $q=r^{f}$ for an integer $f$. Take $s\in(K_{0})^{*}$ such that
$C_{(K_{0})^{*}}(s)=C_{(r^{5}-1)/(r-1)}\times C_{(r^{5}-1)/(r-1)}$ and
$C_{K^{*}}(s)=C_{(q^{5}-1)/(q-1)}\times C_{(q^{5}-1)/(q-1)}$. Then
$B=\mathcal{E}_{5}(K,(s))$ is a $5$-block with defect group $C_{5}\times
C_{5}$. Since $s\in(K_{0})^{*}$, it follows that $\sigma$ stabilises $B$. In
particular, if $p\mid f$, then $B$ is stabilised by a field automorphism of
order $p$.

Similarly we can construct $3$-blocks of some finite exceptional groups that
are stabilised by a field automorphism of order $3$.

$(b)$ Suppose $a=1$. Then $p\nmid f$, and $K$ has no field automorphism of
order $p$ except when $p=3$ and $K={}^{3}D_{4}(q)$, in which case the field
automorphism group of $K$ is cyclic of order $3f$.

$(c)$ Let $p=3$, $K={}^{3}D_{4}(q)$, and let $D\leq K$ be a defect group given
by Proposition 5.2 $(c)$ and $Q\leq K$ a subgroup such that $Q\cong
C_{p}\times C_{p}$ but $Q\neq_{K}D$. By [27, Table 4.7.3A], $K$ has two order
$3$ conjugacy classes, so there is $x\in Q$ such that
$C_{K}(x)=(q-\epsilon)\circ_{2^{*}}({\rm SL}_{2}(q^{3}).2^{*})$. Thus $Q\leq
C_{K}(x)$, $Q=\langle x,y\rangle$ with $y\in{\rm SL}_{2}(q^{3})$ and
$C_{K}(Q)=C_{q-\epsilon}\times C_{q^{3}-\epsilon}$. In particular, $Q$ is not
radical. Therefore, $D$ is the only defect group of $K$ isomorphic to
$C_{p}\times C_{p}$.

Recall $K=O^{r^{\prime}}(C_{{\overline{K}}}(\sigma))$, so $K\neq
C_{{\overline{K}}}(\sigma)$ in general. But if $K=K_{u}$, then
$K=C_{{\overline{K}}}(\sigma)$. Note that if $K$ is given by Proposition 5.2
(b) or (c), then $K=K_{u}$ except when $K=(E_{7})_{a}(q)=(E_{7})_{u}(q)/Z$, in
which case $Z=Z((E_{7})_{u}(q))=C_{2}$. Since $p$ is odd and $Z$ is a
$p^{\prime}$-group, we may always suppose $K=K_{u}$, and hence
$K=C_{{\overline{K}}}(\sigma)$.

Let $B$ be a block of $K$ with a defect group $D$ which is given by
Proposition 5.2 (b) or (c), then $K=K_{u}$ and hence $D$ is toral. In the
following we are going to show that if $B$ is stabilised by a field
automorphism $\sigma^{\prime}$ of order $p$, then $D$ is centralised by
$\sigma^{\prime}$.

Recall that $q=r^{f}$ and $e^{\prime}$ is the multiplicative order of
$\epsilon r$ modulo $p$ and $e$ is the multiplicative order of $\epsilon q$
modulo $p$. Suppose $e=1$. If $r\equiv\pm 1\pmod{5}$, then $e^{\prime}=1$ and
$e^{\prime}\mid f$. If $r\equiv\pm 2\pmod{5}$, then $r^{2}\equiv-1\pmod{5}$,
$e^{\prime}=2$ and $|r|=4$ in $C_{5}$. But $r^{2f}\equiv 1\pmod{5}$, so $4\mid
2f$ and $e^{\prime}=2\mid f$. Write $f=e^{\prime}f_{1}$ and
$q_{0}=r^{e^{\prime}}$. If $p=3$, then $e=e^{\prime}=1$, we set $q_{0}=r$. Let
$K_{0}\leq K$ be the subgroup of $K$ defined over $q_{0}$. So

$K_{0}\in\\{E_{8}(q_{0}),{}^{3}D_{4}(q_{0}),F_{4}(q_{0}),(E_{6})_{u}^{-\epsilon}(q_{0}),(E_{7})_{u}(q_{0})\\}$

with $K_{0}\leq K$. Denote by $({\overline{K}},\sigma_{0})$ a $\sigma$-setup
for $K_{0}$, so $K_{0}=C_{{\overline{K}}}(\sigma_{0})$ and
$\sigma=\sigma_{0}^{f_{1}}$.

Let $\ell$ be the Lie rank of ${\overline{K}}$. Take a maximal torus
$T_{\epsilon}=(q_{0}-\epsilon)^{\ell}$ of $K_{0}$ and so
$E=\Omega_{1}(O_{p}(T_{\epsilon}))=(C_{p})^{\ell}$. Thus
$C_{K_{0}}(E)=T_{\epsilon}$, $C_{K}(E)=T_{K}\leq K$ and
$C_{{\overline{K}}}(E)={\overline{T}}$ are maximal tori of $K$ and
${\overline{K}}$, respectively. Note that

$E=\Omega_{1}(O_{p}(T_{K}))={\overline{T}}_{(p)}:=\\{x\in{\overline{T}}\mid
x^{p}=1\\}.$

Let $V$ be the extended Weyl group of ${\overline{K}}$, so that
$N_{{\overline{K}}}({\overline{T}})=\langle{\overline{T}},V\rangle$,
$V/({\overline{T}}\cap V)$ is a Weyl group and $V$ is centralised by all field
automorphisms of ${\overline{K}}$. In particular, $V\leq K_{0}$.

###### Lemma 5.4

Let $p$ be an odd prime and follow the notation above. Let $K$ be a
quasisimple group and $B\in{\rm Blk}(K)$ with defect group $D=C_{p}\times
C_{p}$ given in Proposition 5.2 $(b)$ or $(c)$. Then $K_{0}$ contains a
$K$-conjugate of $D$.

Proof: Let $X\leq K_{0}$ be a toral elementary abelian $p$-subgroup. By
Proposition 5.1, $X=A_{w}$ is ${\overline{K}}$-conjugate to some
$A\leq{\overline{T}}$, where $w\in
C_{{\overline{K}}}(A)/C_{{\overline{K}}}(A)^{\circ}$. Note that $A\leq
T_{(p)}=E$, so $A\leq K_{0}$. If $x\in N_{{\overline{K}}}(A)$, then
${\overline{T}},{\overline{T}}^{x}\leq C_{\overline{G}}(A)^{\circ}$, so
${\overline{T}}^{xy}={\overline{T}}$ for some $y\in
C_{\overline{G}}(A)^{\circ}$. Thus $xy=vt\in
N_{{\overline{K}}}({\overline{T}})$ with $v\in V,t\in{\overline{T}}$. Hence
$x=vu$ for some $u\in C_{\overline{G}}(A)^{\circ}$,
$N_{{\overline{K}}}(A)=C_{{\overline{K}}}(A)^{\circ}N_{V}(A)$ and
$C_{{\overline{K}}}(A)=C_{{\overline{K}}}(A)^{\circ}C_{V}(A)$. Since $V\leq
K_{0}$, we have that $\sigma_{0}$ induces identity map on
$N_{{\overline{K}}}(A)/C_{{\overline{K}}}(A)^{\circ}$ and
$C_{{\overline{K}}}(A)/C_{{\overline{K}}}(A)^{\circ}$. In particular, a
$\sigma_{0}$-class of $N_{{\overline{K}}}(A)/C_{{\overline{K}}}(A)^{\circ}$
contained in $C_{{\overline{K}}}(A)/C_{{\overline{K}}}(A)^{\circ}$ is a
$N_{{\overline{K}}}(A)/C_{{\overline{K}}}(A)^{\circ}$-conjugacy class
contained in $C_{{\overline{K}}}(A)/C_{{\overline{K}}}(A)^{\circ}$.

Repeat the proof above with $X\leq K_{0}$ replaced by $X\leq K$. We have that
$X=A_{w}$ with $w\in C_{{\overline{K}}}(A)/C_{{\overline{K}}}(A)^{\circ}$,
$A\leq T_{K}$, and $\sigma$ induces identity map on
$N_{{\overline{K}}}(A)/C_{{\overline{K}}}(A)^{\circ}$ and
$C_{{\overline{K}}}(A)/C_{{\overline{K}}}(A)^{\circ}$. Since
$N_{{\overline{K}}}(A)/C_{{\overline{K}}}(A)^{\circ}$ and
$C_{{\overline{K}}}(A)/C_{{\overline{K}}}(A)^{\circ}$ are independent of
$q_{0}$ and $q$, we have that $K$ and $K_{0}$ have the same number of
conjugacy classes of toral elementary abelian $p$-subgroups. In particular,
the $K$-conjugacy class of $D$ contains a representative in $K_{0}$. $\Box$

###### Corollary 5.5

Let $p$ be an odd prime and let $K$ be a quasisimple group and $B\in{\rm
Blk}(K)$ with defect group $D=C_{p}\times C_{p}$ given by Proposition 5.2
$(b)$ or $(c)$. Suppose $\sigma^{\prime}$ is a field automorphism of $K$ of
order $p$. Then we may suppose $\sigma^{\prime}$ centralises $D$.

Proof: Follow the notations of Lemma 5.4. We may suppose $D\leq K_{0}$. Note
that $f=e^{\prime}f_{1}$ with $e^{\prime}\in\\{1,2\\}$. If $p\mid f$, then
$p\mid f_{1}$ and so $f_{1}=pf_{2}$ for some integer $f_{2}$. We may suppose
$\sigma^{\prime}=\sigma_{0}^{f_{2}}$ and hence $\sigma^{\prime}$ centralizes
$K_{0}=C_{{\overline{K}}}(\sigma_{0})$, and in particular, $\sigma^{\prime}$
centralizes $D$.

Suppose $p\nmid f$. If $K\neq{}^{3}D_{4}(q)$, then $K$ has no field
automorphism of order $p$. Suppose $K={}^{3}D_{4}(q)$, so field automorphism
group of $K$ is cyclic of order $3f$, and an order $3$ field automorphism is
also an order $3$ graph automorphism. By [27, Table 4.7.3A]), $K$ has two
order $3$ field automorphisms $\gamma_{1},\gamma_{2}$ such that
$C_{K}(\gamma_{1})=G_{2}(q)$ and $C_{K}(\gamma_{2})={\rm
PSL}_{3}^{\epsilon}(q).3$. In addition, $\gamma_{1}=h\gamma_{2}$ for some
$h\in K$. So $\gamma_{1}$ centralises a $K$-conjugate of $D$ if and only if
$\gamma_{2}$ centralises a $K$-conjugate of $D$. Take $E=C_{p}\times
C_{p}\leq{\rm PSL}_{3}^{\epsilon}(q)$. Then $E$ is Sylow and ${\rm Out}_{{\rm
PSL}_{3}^{\epsilon}(q)}(E)=Q_{8}$. If $Q$ is given in Remark 5.3 (c), then
${\rm Out}_{K}(Q)=D_{12}$ (cf. [2, (1A)]), so $E\neq_{K}Q$ and hence
$E=_{K}D$. In particular, $\sigma^{\prime}$ centralises a $K$-conjugate of
$D$. $\Box$

###### Lemma 5.6

Let $p$ be an odd prime and let $K$ be a quasisimple group of Lie type and
$B\in{\rm Blk}(K)$ with a defect group $D=C_{p}\times C_{p}$ given in
Proposition 5.2 $(b)$ or $(c)$. Suppose $K\leq H$ such that $[H{:}K]=p$ and
$H/K\leq{\rm Out}(K)$. Let $B_{H}$ be a block of $H$ covering $B$. Then a
defect group of $B_{H}$ is abelian of order $p^{2}$ or $p^{3}$.

Proof: Let $(D,b_{D})$ be a Sylow $B$-subgroup. By [16, Lemma 4.1], there is a
$B_{H}$-subgroup $(D,b_{H})$ such that $b_{H}$ covers $b_{D}$.

(a) First we show that $C_{H}(D)$ contains a defect group $D_{H}$ of $B_{H}$.

We may suppose $D\leq D_{H}$ for some $D_{H}=D(B_{H})$ and so $|D_{H}{:}D|\leq
p$. If $D=D_{H}$, then Lemma 5.6 holds. Suppose $|D_{H}{:}D|=p$. Note that a
Sylow $p$-subgroup of ${\rm Out}(K)$ is cyclic, so by Corollary 5.5, we may
suppose $H=\langle K,\sigma^{\prime}\rangle$, where $\sigma^{\prime}$ is an
order $p$ field automorphism such that $[D,\sigma^{\prime}]=1$. Thus
$\sigma^{\prime}\in C_{H}(D)$,

$C_{H}(D)=\langle C_{K}(D),\sigma^{\prime}\rangle$

and $C_{H}(D)/C_{K}(D)\cong C_{p}$. Note that if $y\in N_{H}(D,b_{H})$, then
$y$ stabilises $b_{D}$, so $N_{H}(D,b_{H})\leq N_{H}(D,b_{D})$. But
$\sigma^{\prime}\in C_{H}(D)$, so $\sigma^{\prime}$ stabilizes $b_{D}$. Since

$N_{H}(D,b_{D})/N_{K}(D,b_{D})\leq H/K,$

we have

$N_{H}(D,b_{D})=\langle
N_{K}(D,b_{D}),\sigma^{\prime}\rangle=C_{H}(D)N_{K}(D,b_{D}).$

So $N_{H}(D,b_{D})/C_{H}(D)\cong N_{K}(D,b_{D})/C_{K}(D)$ is a
$p^{\prime}$-group, hence $N_{H}(D,b_{H})/C_{H}(D)$ is a $p^{\prime}$-group.

Since $|D_{H}{:}D|=p$, we have that $(D,b_{H})$ is not a Sylow
$B_{H}$-subgroup and hence $|N_{H}(D,b_{H}){:}D|_{p}=p$. But
$N_{H}(D,b_{H})/C_{H}(D)$ is a $p^{\prime}$-group, so $C_{H}(D)$ contains a
defect group, say $D_{H}$ of $B_{H}$.

Suppose $(K,p)=({}^{3}D_{4}(q),3)$. Then
$C_{K}(D)=T_{2,\epsilon}\cong(C_{q^{2}+\epsilon q+1})^{2}$ as shown in the
proof of Proposition 5.2 (c). Thus $C_{H}(D)=T_{2,\epsilon}.\sigma^{\prime}$
and $\langle D,\sigma^{\prime}\rangle$ is an abelian Sylow $p$ subgroup of
$C_{H}(D)$, Hence $D_{H}\leq C_{H}(D)$ is abelian.

Suppose

$(K,p)\in\\{(E_{8}(q),5),(F_{4}(q),3),((E_{6})_{u}^{-\epsilon}(q),3),(E_{8}(q),3),((E_{7})_{u}(q)/Z,3)\\}.$

As shown in the proof of Proposition 5.2 (b) or (c),

$C_{K}(D)=C_{K}(D)^{\circ}.w=C_{K}(D)^{\circ}.p,$

where $C_{K}(D)^{\circ}=(C_{{\overline{K}}}(D)^{\circ})^{\sigma}$ and $w$
induces an order $p$ action on $C_{K}(D)^{\circ}$. Suppose
$K={\overline{K}}^{\sigma}$. Note that ${\overline{K}}$ is simply-connected
(or universal), so every elementary abelian subgroup of size $p^{2}$ is a
subgroup of a maximal torus. We may suppose
$D\leq{\overline{T}}\leq{\overline{K}}$ for some maximal torus
${\overline{T}}$. So ${\overline{T}}\leq C_{{\overline{K}}}(D)^{\circ}$. Since
$w\not\in C_{K}(D)^{\circ}=(C_{{\overline{K}}}(D)^{\circ})^{\sigma}$, we have
$w\in C_{{\overline{K}}}(D)\setminus C_{{\overline{K}}}(D)^{\circ}$.

(b) Next we show that there is an element $v\in C_{K}(D)\setminus
C_{K}(D)^{\circ}$ such that $[v,\sigma^{\prime}]=1$. (This is well-konwn and
proved for all toral elementary abealin $p$-subgroups of ${\overline{K}}$ in
[3], and we give a proof here for completeness).

Since $w\in C_{{\overline{K}}}(D)\setminus C_{{\overline{K}}}(D)^{\circ}$, we
have ${\overline{T}}$ and ${\overline{T}}^{w}$ are two maximal tori of
$C_{{\overline{K}}}(D)^{\circ}$, and uniqueness implies that
${\overline{T}}^{wc}={\overline{T}}$ for some $c\in
C_{{\overline{K}}}(D)^{\circ}$. Thus $wc\in
N_{{\overline{K}}}({\overline{T}})={\overline{T}}V$ and $wct\in V\leq K_{0}$
for some $t\in{\overline{T}}$, where $V$ is the extended Weyl group (cf. Lemma
5.4). Set $v=wct$, so $v\in C_{K}(D)$ and $v\not\in C_{K}(D)^{\circ}$ as
$v\not\in C_{{\overline{K}}}(D)^{\circ}$. Since
$C_{K}(D)/C_{K}(D)^{\circ}\cong C_{p}$, we have $C_{K}(D)=C_{K}(D)^{\circ}.v$.

(c) Now we show that $D_{H}$ is abelian.

Suppose

$(K,p)\in\\{(E_{8}(q),5),(F_{4}(q),3),((E_{6})_{u}^{-\epsilon}(q),3)\\}.$

As shown in the proof of of Proposition 5.2 (b) or (c),
$C_{K}(D)\cong(C_{q^{4}+\epsilon q^{3}+q^{2}+\epsilon q+1})^{2}.5$ when
$(K,p)\in(E_{8}(q),5)$; $C_{K}(D)\cong(C_{q^{2}+\epsilon q+1})^{2}.3$ when
$(K,p)\in(F_{4}(q),3)$; $C_{K}(D)=(C_{q^{2}+\epsilon q+1}\times
C_{q^{4}+q^{2}+1}).3$ when $(K,p)=((E_{6})_{u}^{-\epsilon}(q),3)$. Thus
$C_{K}(D)=T.p$, where $C_{K}(D)^{\circ}=T$ is a maximal torus of $K$. By the
proof of part (b) above, $C_{K}(D)=\langle T,v\rangle$ for some $v\not\in T$
and $[v,\sigma^{\prime}]=1$.

Since $v^{p}\in T$ and $v\not\in T$, we have that the abelian subgroup
$\langle D,v,\sigma^{\prime}\rangle$ contains a Sylow $p$-subgroup of
$C_{H}(D)$. Hence any Sylow $p$-subgroup of $C_{H}(D)$ is abelian, so is
$D_{H}$.

Suppose $(K,p)\in\\{(E_{8}(q),3),((E_{7})_{u}(q)/Z,3)\\}$. By Proposition 5.2
(c),

$C_{K}(D)=(C_{q^{2}+\epsilon q+1}\times C_{q^{2}+\epsilon q+1}\times
L).v\quad\hbox{ and }\quad C_{H}(D)=\langle C_{K}(D),\sigma^{\prime}\rangle,$

where $v$ acts as an order $p$ element on $C_{K}(D)^{\circ}$,
$[v,\sigma^{\prime}]=1$ and $L=({\rm SL}_{2}(q^{3})/Z)$ or ${}^{3}D_{4}(q)$
according as $K=(E_{7})_{u}(q)/Z$ or $E_{8}(q)$. Note that $L$ is a normal
subgroup of $C_{H}(D)$. If $D_{H}\cap L\neq 1$, then $(D_{H}\cap L)\cong
C_{p}$ and $D_{H}=D\times(D_{H}\cap L)$ is abelian. Suppose $D_{H}\cap L=1$,
so that $D_{H}\cong(D_{H}L)/L\leq C_{H}(D)/L=\langle
C_{K}(D)/L,\sigma^{\prime}\rangle$. Note that the abelian subgroup $\langle
D,v,\sigma^{\prime}\rangle L/L$ contains a Sylow $p$-subgroup of $C_{H}(D)/L$,
hence any Sylow $p$-subgroup of $C_{H}(D)/L$ is abelian, so is $D_{H}$.

$\Box$

##  6 Reductions

The first step of our reduction is to apply Fong-Reynolds and Külshammer-Puig
reductions as far as possible, meaning that we may establish a Morita
equivalence with what we will call a reduced block. A feature of these
particular Morita equivalences is that they preserve both fusion and the
Külshammer-Puig classes as reviewed in Section 2. The proof of this part of
the reduction is based on the first part of the proof of [18, Proposition
4.3]. Once we have established that it suffices to consider reduced blocks, we
will then analyse their structure.

###### Proposition 6.1

Let $G$ be a finite group and let $B$ be a block of ${\mathcal{O}}G$ with
defect group $D$. There is a finite group $H$ and a block $C$ of
${\mathcal{O}}H$ with defect group isomorphic to $D$ such that $B$ is basic
Morita equivalent to $C$ and the following are satisfied:

  1. (R1)

$C$ is quasiprimitive, that is, every covered block of a normal subgroup of
$H$ is $H$-stable;

  2. (R2)

If $N\lhd H$ and $C$ covers a nilpotent block of ${\mathcal{O}}N$, then $N\leq
Z(H)O_{p}(H)$ and $O_{p^{\prime}}(N)\leq Z(H)\cap H^{\prime}$.

The block $C$ has the same fusion system $\mathcal{F}$ as $B$, and the same
Külshammer-Puig class in $H^{2}({\rm Aut}_{\mathcal{F}}(D),k^{\times})$.

Proof: We use two process, labelled (FR) and (KP), which we apply alternately
to blocks with defect groups isomorphic to $D$ and their normal subgroups.
Each process strictly decreases $[G:O_{p^{\prime}}(Z(G))]$ when applied
nontrivially and so repeated application must come to an end, resulting in a
block satisfying (R1) and (R2).

Let $G$ be a finite group and $B$ be a block of ${\mathcal{O}}G$ with defect
group $D$.

  1. (FR)

Suppose $N\lhd G$ and $b$ is a block of ${\mathcal{O}}N$ covered by $B$. Write
$I=I_{G}(b)$ for the stabilizer of $b$ in $G$, and $B_{I}$ for the Fong-
Reynolds correspondent, which is the unique block of $I$ covering $b$ with
Brauer correspondent $B$. Suppose that $I\neq G$. Now by [31, Theorem 6.8.13]
$B_{I}$ is source algebra equivalent to $B$. Hence by [31, Theorem 8.7.1]
$B_{I}$ has defect group $D$ and the same fusion system, and by [31, Theorem
8.14.1] it has the same Külshammer-Puig class. Since $O_{p^{\prime}}(Z(G))\leq
O_{p^{\prime}}(Z(I))$ and $I\neq G$, we have
$[I:O_{p^{\prime}}(Z(I))]<[G:O_{p^{\prime}}(Z(G))]$. Process (FR) involves
replacing $B$ by $B_{I}$ and repeating the process for all normal subgroups
until $B$ is quasiprimitive.

  2. (KP)

Assume that process (FR) has been performed, so that $B$ is quasiprimitive.
Let $N\lhd G$ such that $N\not\leq(Z(G)\cap[G,G])O_{p}(G)$, and suppose that
$B$ covers a nilpotent block $b$ of $N$. Let $b^{\prime}$ be a block of
$Z(G)N$ covered by $B$ and covering $b$. Since $B$ is quasiprimitive, both $b$
and $b^{\prime}$ are $G$-stable. Further $b^{\prime}$ must also be nilpotent.
By [31, Theorem 8.12.5] and [31, Remark 8.12.8] $B$ is basic Morita equivalent
to a block $\tilde{B}$ of a central extension $\tilde{L}$ of a finite group
$L$ by a $p^{\prime}$-group $\tilde{Z}$ (which we may further assume is
contained in the derived subgroup $[\tilde{L},\tilde{L}]$) such that there is
$\tilde{M}\lhd\tilde{L}$ with $\tilde{M}\cong D\cap(Z(G)N)$,
$G/Z(G)N\cong\tilde{L}/\tilde{Z}\tilde{M}$, and $\tilde{B}$ has defect group
isomorphic to $D$, the same fusion system as $B$, and the same Külshammer-Puig
class. We have
$[\tilde{L}:O_{p^{\prime}}(Z(\tilde{L}))]\leq|L|=[G:Z(G)N]|D\cap(Z(G)N)|<[G:O_{p^{\prime}}(Z(G))]$
and $\tilde{M}\leq(Z(\tilde{L})\cap[\tilde{L},\tilde{L}])O_{p}(\tilde{L})$.
Process (KP) consists of replacing $G$ by $\tilde{L}$ and $B$ by $\tilde{B}$.

Repeated application of (FR) and (KP) to all blocks of all normal subgroups
must final result in a block to which we may no longer apply (FR) or (KP),
that is, a block $C$ satisfying (R1) and (R2). $\Box$

Call a pair $(G,B)$ satisfying conditions (R1)-(R2) of the proposition
_reduced_. If the group $G$ is clear, then just say that $B$ is reduced.

Reduced blocks with defect group $D\cong p_{+}^{1+2}$ have a very restricted
structure:

###### Proposition 6.2

Let $(G,B)$ be a reduced pair where $D$ has defect group $D\cong p_{+}^{1+2}$
with $p\geq 3$. Then either: (i) $B$ is a block of maximal defect and one of
the following occurs:

  1. (S1)

$D\lhd G$;

  2. (S2)

$G\cong(C_{p}\times C_{p})\rtimes H$, where $SL_{2}(p)\leq H\leq GL_{2}(p)$;

  3. (S3)

$G$ satisfies $PSL_{3}(p)\leq G/Z(G)\leq{\rm Aut}(PSL_{3}(p))$, where
$Z(G)\leq G^{\prime}$;

  4. (S4)

$G$ satisfies $PSU_{3}(p)\leq G/Z(G)\leq{\rm Aut}(PSU_{3}(p))$, where
$Z(G)\leq G^{\prime}$;

  5. (S5)

$p=3$ and $G/Z(G)$ is one of $M_{12}$, $M_{12}.2$, $M_{24}$, $J_{2}$,
$J_{2}.2$, $J_{4}$, $He$, $He.2$ or $Ru$;

  6. (S6)

$p=5$ and $G/Z(G)$ is one of $2.HS$, $2.HS.2$, $3.McL$, $McL.2$, $2.Ru$,
$Co_{3}$, $Co_{2}$ or $Th$;

  7. (S7)

$p=11$ and $G=J_{4}$, and $B$ is the principal block;

  8. (S8)

$p=13$ and $G$ is the Fischer-Griess Monster $M$, and $B$ is the principal
block;

or (ii) $p=3$ and one of the following occurs:

  1. (S9)

$G$ is $Co_{1}$ or the Fischer-Griess Monster $M$;

  2. (S10)

$G$ satisfies $PSL_{3\delta}(q)\leq G/Z(G)\leq{\rm Aut}(PSL_{3\delta}(q))$
with $|Z(G)|_{3}=3$, where $3\parallel q-1$ and $Z(G)\leq G^{\prime}$;

  3. (S11)

$G$ satisfies $PSU_{3\delta}(q)\leq G/Z(G)\leq{\rm Aut}(PSU_{3\delta}(q))$
with $|Z(G)|_{3}=3$, where $3\parallel q+1$ and $Z(G)\leq G^{\prime}$;

  4. (S12)

$G$ satisfies $G_{2}(q)\leq G\leq{\rm Aut}(G_{2}(q))$ for some $q$ not
divisible by $3$, and $B$ is the principal block;

  5. (S13)

$G$ satisfies ${}^{2}F_{4}(2^{2m+1})\leq G\leq{\rm
Aut}({}^{2}F_{4}(2^{2m+1}))$ for some $m\in\mathbb{N}$, and $B$ is the
principal block.

Proof: Consider first $O_{p}(G)$, which is a subgroup of $D$. We may suppose
that $O_{p}(G)\neq D$ as otherwise we are in case (S1).

Suppose that $|O_{p}(G)|=p^{2}$. Now the unique block of
${\mathcal{O}}C_{G}(O_{p}(G))$ covered by $B$ has central defect group $D\cap
C_{G}(O_{p}(G))=O_{p}(G)$. Hence it is nilpotent and by (R2)
$C_{G}(O_{p}(G))=O_{p}(G)Z(G)$. Hence $B$ has maximal defect. Now
$O_{p}(G)\cong C_{p}\times C_{p}$, so $G/C_{G}(O_{p}(G))$ is isomorphic to a
subgroup of $GL_{2}(p)$ containing a Sylow $p$-subgroup, acting on $O_{p}(G)$
by the natural representation. Since we do not have $D\lhd G$, by Lemma 2.2
this means $SL_{2}(p)\leq G/C_{G}(O_{p}(G))\leq GL_{2}(p)$. Since $B$ is
reduced we have $O_{p^{\prime}}(G)\leq Z(G)\cap[G,G]$. The Schur multiplier of
$G/O_{p^{\prime}}(G)$ has trivial $p^{\prime}$-part, so $O_{p^{\prime}}(G)=1$
and $C_{G}(O_{p}(G))=O_{p}(G)$. Hence by Lemma 2.3 we are in case (S2).

We assume from now on that either $|O_{p}(G)|=p$, in which case
$O_{p}(G)=Z(D)$, being the unique normal subgroup of $D$ of order $p$, or
$O_{p}(G)=1$.

Now consider $E(G)$, the layer of $G$, which is the product of the components
of $G$. The generalized Fitting subgroup is $F^{*}(G)=E(G)F(G)$, where $F(G)$
is the Fitting subgroup, the product of the groups $O_{l}(G)$ for primes $l$.
See [7] for details. For reduced pairs we have $F(G)=O_{p}(G)Z(G)$. Note that
by [7, 31.13] $C_{G}(F^{*}(G))\leq F^{*}(G)$. If $E(G)=1$, then
$G/Z(G)O_{p}(G)$ is isomorphic to a subgroup of the $p^{\prime}$-group ${\rm
Aut}(C_{p})$, so that $O_{p}(G)\in{\rm Syl}_{p}(G)$, contradicting our
assumption that $O_{p}(G)\neq D$. Hence $E(G)\neq 1$. Write
$E(G)=L_{1}\circ\cdots\circ L_{t}$, where the $L_{i}$ are the components of
$G$. Since the $p$-rank of a quotient of a subgroup of $D$ is at most two, it
follows from (R2) that $t\leq 2$ (see for example the proof of [19, Theorem
3.2] where a similar argument is given in full detail).

Suppose that $t=2$, so that $|D\cap E(G)|\geq p^{2}$. Note that $D$ stabilizes
$L_{1}$ and $L_{2}$. We cannot have $|D\cap E(G)|=p^{2}$, since then $D$ would
have distinct normal subgroups $L_{1}\cap D$ and $L_{2}\cap D$ of order $p$,
which cannot happen in $p^{1+2}$. Hence $D\leq E(G)$, $O_{p}(G)\leq Z(E(G))$
and $|L_{i}\cap D|=p^{2}$ for each $i$. However this would mean that $D$ is
abelian, a contradiction. Hence $t=1$, so $L_{1}/Z(L_{1})\leq
G/Z(G)O_{p}(G)\leq{\rm Aut}(L_{1}/Z(L_{1}))$.

When $D\leq L_{1}$, we apply [5] as follows. If $L_{1}/Z(L_{1})$ is sporadic,
then it follows from [5, Table 1] that $G$ is as in (S5)-(S9). By Section 2 of
[5] there are no blocks of double covers of alternating groups with defect
group $D$. If $L_{1}$ is of Lie type (in defining or non-defining
characteristic), then [5, Theorem 4.5] (for classical groups), [5, Theorem
5.1] (for exceptional groups) and the discussion following the proof of [5,
Theorem 5.1] (for exceptional covers of simple groups) tells us that we are in
one of cases (S3), (S4) or (S10)-(S13).

Hence we may suppose that $D\cap L_{1}$ has order dividing $p^{2}$. Note that
we must have $O_{p}(G)\leq L_{1}$, otherwise $D$ has distinct normal subgroups
$O_{p}(G)$ and $D\cap L_{1}$, a contradiction. We must have that ${\rm
Out}(L_{1}/Z(L_{1}))$ has order divisible by $p$, otherwise $[G:L_{1}]$ would
have order prime to $p$, contradicting our assumption that $[D:D\cap L_{1}]=p$
or $p^{2}$.

If $L_{1}$ is a finite group of Lie type defined over a field of
characteristic $p$, then $D\cap L_{1}\in{\rm Syl}_{p}(L_{1})$, so that $L_{1}$
is $PSL_{2}(p^{2})$ or $SL_{2}(p^{2})$. But then ${\rm Out}(PSL_{2}(p^{2}))$
has order prime to $p$, a contradiction. If $L_{1}$ is a sporadic or
alternating group, then we have that ${\rm Out}(L_{1}/Z(L_{1}))$ has order $1$
or $2$, a contradiction.

Suppose that $L_{1}$ is a finite group of Lie type defined over a field of
characteristic different to $p$. We consider the structure of the outer
automorphism group, found in [13, Table 5].

From Proposition 4.2, if $L_{1}$ is classical but not special linear, unitary
or type $D_{4}$, then it has outer automorphism group of order prime to $p$,
so that $G/L_{1}$ is a $p^{\prime}$-group, a contradiction. If $L_{1}$ is of
type $D_{4}$, then Proposition 4.2 (c) tells us that $D$ is not a defect group
of $G$.

Suppose $L_{1}$ is special linear or unitary. Since the symmetry of the Dynkin
diagram has order at most $2$ and the diagonal automorphisms contribute a
normal cyclic subgroup to the outer automorphism group, it follows that $G$
has a normal subgroup $N$ such that $[N{:}L_{1}]\mid p$ and $N$ acts on
$L_{1}$ as inner-diagonal automorphisms. By Proposition 4.1 either there is no
field automorphism of order $p$ or $D\cap N$ is cyclic. In the former case,
$N$ has a $p^{\prime}$-index in $G$, so that $D\leq N$. But [5, Proposition
4.1] tells us that (since $D\cap L_{1}$ is abelian) $D$ must be abelian, a
contradiction. In the later case, $G/N$ has a cyclic Sylow $p$-subgroup. Since
$D\cap N$ is cyclic and normal in $D$, it follows that $D\cap N=Z(D)$ and
$D/(D\cap N)\cong C_{p}\times C_{p}$, which is impossible.

Suppose that $L_{1}$ is of exceptional type. By Proposition 5.2 unless $L_{1}$
is as in (b) or (c) of Proposition 5.2, there are no field automorphisms of
order divisible by $p$, and further the outer automorphism group of $L_{1}$ is
prime to $p$, a contradiction.

Finally suppose that $L_{1}$ is as in one of the cases within (b) and (c) of
Proposition 5.2. Note that in all these cases ${\rm Out}(L_{1})$ has a normal
Sylow $p$-subgroup, which must necessarily have order $3$ since $B$ is
reduced. Let $H$ be the preimage of $O_{p}(G/L_{1})$ in $G$. Then $D\leq H$
and $D$ is abelian by Lemma 5.6, a contradiction. $\Box$

Proof of Theorem 1.1 Let $G$ be a finite group and $B$ a block of
${\mathcal{O}}G$ with defect group $D\cong p_{+}^{1+2}$ where $p\geq 5$. By
Proposition 6.1 we may assume that $(G,B)$ is reduced. By Proposition 6.2 and
Section 2 $B$ is one of the blocks listed. $\Box$

Proof of Theorem 1.4 By Proposition 6.1, it suffices to show that there are
only finitely many Morita equivalence classes amongst the reduced blocks with
defect group $3_{+}^{1+2}$. By [21, Corollary 4.18], it further suffices to
show that there are only finitely many Morita equivalence classes amongst the
unique blocks of ${\mathcal{O}}\langle D^{g}:g\in G\rangle$ covered by reduced
blocks $B$. Examining the list given in Proposition 6.2 this means that we may
assume one of the following:

(i) $G=D$;

(ii) $G\cong(C_{3}\times C_{3})\rtimes SL_{2}(3)$;

(iii) $G$ is quasisimple.

In cases (i) and (ii) there are clearly only finitely many possible Morita
equivalence classes, so it suffices to assume that $G$ is quasisimple.

Following [18], the ${\mathcal{O}}$-Morita Frobenius number ${\rm
mf}_{\mathcal{O}}(B)$ of a block $B$ is defined to be the smallest integer $n$
such that $B^{(p^{n})}$ is Morita equivalent to $B$, where $B^{(p^{n})}$ is
defined as follows. Define a ring automorphism $\bar{\sigma}:kG\rightarrow kG$
by $\bar{\sigma}(\sum_{g\in G}a_{g}g)=\sum_{g\in G}(a_{g})^{p}g$. Write
$B^{(p^{n})}$ for the lift of the image of $k\otimes B$ under
$\bar{\sigma}^{n}$. By [18, Corollary 3.11], in order to show that there are
only finitely many Morita equivalence classes amongst blocks of quasisimple
groups with defect group $3_{+}^{1+2}$, it suffices to show that there
integers $c$ and $m$ such that for all blocks $B$ of quasisimple groups with
defect group $3_{+}^{1+2}$ we have ${\rm c}(B)\leq c$ and ${\rm
mf}_{\mathcal{O}}(B)\leq m$. By [22] we may take $m=4$, and we are done.

##  7 Blocks with defect group $5_{+}^{1+2}$

We apply Theorem 1.1 in the case $p=5$, and classify the Morita equivalence
classes. Note that it may in theory be possible that these Morita equivalence
classes contain blocks with defect group $5_{-}^{1+2}$, although we do not
know of such an example.

Referring to Section 2, the possible inertial quotients correspondent to
conjugacy classes of subgroups of $GL_{2}(5)$ of order prime $5$, of which
there are thirty, obtained by a routine calculation in MAGMA [9]. Each block
with normal defect group is Morita equivalent to a block of
$5_{+}^{1+2}\rtimes\hat{E}$ where $E$ is the inertial quotient and $\hat{E}$
is a central extension of $E$ by $Z\leq Z(\hat{E})\cap\hat{E}^{\prime}$ acting
trivially on $5_{+}^{1+2}$. In order to describe all blocks with normal defect
group, for each $E$ we choose a central extension with $Z\cong
H^{2}(E,k^{\times})$, which we obtain via the pMultiplicator function in
MAGMA. To differentiate between different actions of isomorphic groups $E$ on
$D=5_{+}^{1+2}$ we use the notation $(E)_{i}$. In our case all $D\rtimes E$
where there is such an ambiguity have order 2000 or less, and so we specify
the action by identifying $D\rtimes E$ in the SmallGroup library of MAGMA (or
equivalently GAP [8]). Table 1 gives each of the thirty possibilities, along
with: SmallGroup representation of $5_{+}^{1+2}\rtimes E$ (when this is at
most 2000); the isomorphism type of $H^{2}(E,k^{\times})$; a candidate for
$\hat{E}$ given by its SmallGroup labelling; and the number of Morita
equivalence classes amongst the blocks of $5_{+}^{1+2}\rtimes\hat{E}$. Note
that when $|H^{2}(E,k^{\times})|=2$ it is easy to verify that the two blocks
of $5_{+}^{1+2}\rtimes\hat{E}$ are in distinct Morita equivalence classes by
checking, using the given SmallGroup representation of $\hat{E}$, that the
number of simple modules in each block is different. There are two cases,
treated in Lemma 7.1 and 7.2, that have larger Schur multipliers and require
us to establish Morita equivalences between nonprincipal blocks.

In Table 2 we give the numbers of irreducible characters of height zero
$k_{0}(B)$, height one $k_{1}(B)$, and the number of irreducible Brauer
characters $l(B)$ for each Morita equivalence class of blocks with normal
defect group. Recall that the height of an irreducible character $\chi$ in a
block with defect group $D$ is the non-negative integer $h$ such that
$\chi(1)_{p}=p^{h}[G:D]_{p}$.

We now make some comments on the calculation of $k_{0}(B)$, $k_{1}(B)$ and
$l(B)$ in Table 2. Where $|D\rtimes\hat{E}|\leq 2000$ we used a SmallGroup
representation of the group and computed character tables using GAP.
Otherwise, using GAP we have formed the holomorph of $5_{+}^{1+2}$ and found
$D\rtimes E$ through a search of its subgroups. We computed the inertia
subgroups of the nontrivial irreducible characters of $D$ in $E$, which in all
cases were groups with trivial Schur multiplier. From this we could deduce the
number of irreducible characters in each block.

$E$ |  $D\rtimes E$ as | $H^{2}(E,k^{\times})$ |  $\hat{E}$ as |  $\\#$Morita eq. | Notes
---|---|---|---|---|---
| SmallGroup |  | SmallGroup | classes |
$1$ | (125,3) | 1 | (1,1) | 1 |
$(C_{2})_{1}$ | (250,5) | 1 | (10,2) | 1 |
$(C_{2})_{2}$ | (250,8) | 1 | (2,1) | 1 |
$C_{3}$ | (375,2) | 1 | (3,1) | 1 |
$(C_{4})_{1}$ | (500,17) | 1 | (4,1) | 1 |
$(C_{4})_{2}$ | (500,21) | 1 | (4,1) | 1 |
$(C_{4})_{3}$ | (500,23) | 1 | (4,1) | 1 |
$(C_{4})_{4}$ | (500,25) | 1 | (4,1) | 1 |
$C_{2}\times C_{2}$ | (500,27) | $C_{2}$ | (8,4) | 2 |
$C_{6}$ | (750,6) | 1 | (6,2) | 1 |
$S_{3}$ | (750,5) | 1 | (6,1) | 1 |
$C_{8}$ | (1000,86) | 1 | (8,1) | 1 |
$(C_{4}\times C_{2})_{1}$ | (1000,89) | $C_{2}$ | (16,6) | 2 |
$(C_{4}\times C_{2})_{2}$ | (1000,91) | $C_{2}$ | (16,6) | 2 |
$D_{8}$ | (1000,92) | $C_{2}$ | (16,7) | 2 |
$Q_{8}$ | (1000,93) | 1 | (8,4) | 1 |
$C_{3}\rtimes C_{4}$ | (1500,35) | 1 | (12,1) | 1 |
$C_{12}$ | (1500,36) | 1 | (12,2) | 1 |
$D_{12}$ | (1500,37) | $C_{2}$ | (24,6) | 2 |
$C_{4}\times C_{4}$ | (2000,473) | $C_{4}$ | (64,18) | 3 | Lemma 7.1
$M_{4}(2)$ | (2000,474) | 1 | (16,6) | 1 |
$C_{4}\circ D_{8}$ | (2000,475) | $C_{2}\times C_{2}$ | (64,74) | 2 | Lemma 7.2
$C_{3}\rtimes C_{8}$ | N/A | 1 | (24,1) | 1 |
$C_{24}$ | N/A | 1 | (24,2) | 1 |
$SL_{2}(3)$ | N/A | 1 | (24,3) | 1 |
$C_{4}\times S_{3}$ | N/A | $C_{2}$ | (48,5) | 2 |
$C_{4}\wr C_{2}$ | N/A | $C_{2}$ | (64,10) | 2 |
$C_{8}\rtimes S_{3}$ | N/A | 1 | (48,5) | 1 |
$C_{4}\circ SL_{2}(3)$ | N/A | 1 | (48,33) | 1 |
$U_{2}(3)$ | N/A | 1 | (96,67) | 1 |
Table 1: Inertial quotients for blocks with defect group $D\cong 5_{+}^{1+2}$ $E$ | $k_{0}(B)$ | $k_{1}(B)$ | $l(B)$
---|---|---|---
$1$ | 20 | 4 | 1
$(C_{2})_{1}$ | 20 | 2 | 2
$(C_{2})_{2}$ | 14 | 8 | 2
$C_{3}$ | 11 | 12 | 3
$(C_{4})_{1}$ | 25 | 1 | 4
$(C_{4})_{2}$ | 13 | 1 | 4
$(C_{4})_{3}$ | 10 | 4 | 4
$(C_{4})_{4}$ | 10 | 16 | 4
$C_{2}\times C_{2}$ | 16,13 | 4,4 | 4,1
$C_{6}$ | 10 | 24 | 6
$S_{3}$ | 13 | 6 | 3
$C_{8}$ | 11 | 2 | 8
$(C_{4}\times C_{2})_{1}$ | 20,14 | 2,2 | 8,2
$(C_{4}\times C_{2})_{2}$ | 14,8 | 8,8 | 8,2
$D_{8}$ | 14,11 | 8,8 | 5,2
$Q_{8}$ | 8 | 20 | 5
$C_{3}\rtimes C_{4}$ | 8 | 24 | 6
$C_{12}$ | 14 | 12 | 12
$D_{12}$ | 14,11 | 12,12 | 6,3
$C_{4}\times C_{4}$ | 25,10,13 | 4,4,4 | 16,1,4
$M_{4}(2)$ | 13 | 4 | 10
$C_{4}\circ D_{8}$ | 16,10 | 10,10 | 10,4
$C_{3}\rtimes C_{8}$ | 13 | 6 | 12
$C_{24}$ | 25 | 6 | 24
$SL_{2}(3)$ | 8 | 28 | 7
$C_{4}\times S_{3}$ | 16,10 | 12,12 | 12,6
$C_{4}\wr C_{2}$ | 20,11 | 5,5 | 14,5
$C_{8}\rtimes S_{3}$ | 20 | 6 | 18
$C_{4}\circ SL_{2}(3)$ | 16 | 14 | 14
$U_{2}(3)$ | 20 | 7 | 16
Table 2: Morita equivalence classes of blocks with defect group $5_{+}^{1+2}$:
blocks of groups with a normal defect group

###### Lemma 7.1

There are precisely three Morita equivalence classes of blocks with normal
defect group $D\cong 5_{+}^{1+2}$ and inertial quotient $C_{4}\times C_{4}$.

Proof: We may choose a central extension $\hat{E}$ of $C_{4}\times C_{4}$ by
$Z\cong C_{4}$ identified as SmallGroup(64,18). Write ${\rm
Irr}(Z)=\\{1_{Z}=\lambda^{0},\lambda,\lambda^{2},\lambda^{3}\\}$, so that
$\lambda^{2}$ is non-faithful, and write $B_{i}$ for the block of
$D\rtimes\hat{E}$ covering $\lambda^{i}\in{\rm Irr}(Z)$. Calculations using
GAP give us that $l(B_{0})=16$, $l(B_{1})=l(B_{3})=1$ and $l(B_{2})=4$, so
that $B_{0}$, $B_{1}$ and $B_{2}$ lie in different Morita equivalence classes.
Now we may embed $D\rtimes\hat{E}$ in $D\rtimes H$, where
$H\cong(C_{2})^{3}.D_{16}$, identified as SmallGroup(128,71), in which
$\lambda$ and $\lambda^{3}$ are permuted. Hence $B_{1}$ and $B_{3}$ are Morita
equivalent. $\Box$

###### Lemma 7.2

There are precisely two Morita equivalence classes of blocks with normal
defect group $D\cong 5_{+}^{1+2}$ and inertial quotient $C_{4}\circ D_{8}$.

Proof: We may choose a central extension $\hat{E}$ of $C_{4}\circ D_{8}$ by
$Z\cong C_{2}\times C_{2}$ identified as SmallGroup(64,73) and embed it as a
normal subgroup in a group $H$ identified as SmallGroup(192,194), in which $Z$
is a noncentral but normal subgroup. Note that $H/Z\cong C_{4}.A_{4}$, which
is SmallGroup(48,33) and is a subgroup of $GL_{2}(5)$, so $H$ acts on $D$ with
kernel $Z$. Hence we may embed $D\rtimes\hat{E}\leq D\rtimes H$. The blocks of
$D\rtimes\hat{E}$ correspond to the irreducible characters of $Z$.
Calculations using GAP show that $H$ permutes the non-trivial irreducible
characters of $Z$, and so the three non-principal blocks of $D\rtimes\hat{E}$
are Morita equivalent. $\Box$

Regarding blocks with non-normal defect group, using [25] and [10] we may
obtain decomposition matrices for all blocks identified in Theorem 1.1(b)-(e)
aside from $Th$. One may check that all these decomposition matrices are
(often subtly) distinct, and further cannot be the decomposition matrix for a
block with normal defect group since there is always an irreducible Brauer
character that does not lift to an ordinary irreducible character. In the case
of $Th$, where no decomposition matrix is currently available, we may use its
character table and the fact that all nonprincipal $5$-blocks have defect zero
to deduce $k(B_{0}(Th))$ and $l(B_{0}(Th))$ (see [13] and [37]). We see that
no other block with this defect group has the same size decomposition matrix.
Table 3 lists the Morita equivalence classes we obtain from blocks with non-
normal defect groups. A subtlety in the definition of the groups in cases (c)
and (d) of Theorem 1.1 is that we are taking a central extension of an
automorphism group of a simple group, and so the Schur multiplier may in
general be different to that of the simple group itself. However, ${\rm
Out}(PSL_{3}(5))$ and ${\rm Out}(PSU_{3}(5))$ are both cyclic, so that by, for
example, [15, Lemma 3.4(ii)] the Schur multiplier of $X$ for $PSL_{3}(5)\leq
X\leq{\rm Aut}(PSL_{3}(5))$ or $PSU_{3}(5)\leq X\leq{\rm Aut}(PSU_{3}(5))$ is
a homomorphic image of that for $PSL_{3}(5)$ or $PSU_{3}(5)$ respectively.

Representative $B$ | $k_{0}(B)$ | $k_{1}(B)$ | $l(B)$ | Inertial quotient | Notes
---|---|---|---|---|---
$(C_{5})^{2}\rtimes SL_{2}(5)$ | 13 | 1 | 5 | $(C_{4})_{2}$ |
$(C_{5})^{2}\rtimes SL_{2}(5).2$ | 20 | 2 | 10 | $(C_{4}\times C_{2})_{1}$ |
$(C_{5})^{2}\rtimes GL_{2}(5)$ | 25 | 4 | 20 | $C_{4}\times C_{4}$ |
$B_{0}(SL_{3}(5))$ | 25 | 4 | 24 | $C_{4}\times C_{4}$ |
$B_{0}(SL_{3}(5).2)$ | 20 | 5 | 18 | $C_{4}\wr C_{2}$ |
$B_{0}(SU_{3}(5))$ | 11 | 2 | 8 | $C_{8}$ |
$B_{1}(SU_{3}(5))$ | 11 | 2 | 8 | $C_{8}$ | Remark 7.3(i)
$B_{0}(SU_{3}(5).2)$ | 13 | 4 | 10 | $M_{4}(2)$ |
$B_{0}(SU_{3}(5).3)$ | 25 | 6 | 24 | $C_{24}$ |
$B_{1}(SU_{3}(5).3)$ | 25 | 6 | 24 | $C_{24}$ | Remark 7.3(ii)
$B_{0}(HS)$ | 13 | 4 | 10 | $M_{4}(2)$ |
$B_{1}(2.HS)$ | 13 | 4 | 10 | $M_{4}(2)$ |
$B_{0}(HS.2)$ | 20 | 5 | 14 | $C_{4}\wr C_{2}$ |
$B_{1}(2.HS.2)$ | 11 | 5 | 5 | $C_{4}\wr C_{2}$ |
$B_{0}(McL)$ | 13 | 6 | 12 | $C_{3}\rtimes C_{8}$ |
$B_{1}(3.McL)$ | 13 | 6 | 12 | $C_{3}\rtimes C_{8}$ | Remark 7.3(iii)
$B_{0}(McL.2)$ | 20 | 6 | 18 | $C_{8}\rtimes S_{3}$ |
$B_{0}(Ru)$ | 20 | 5 | 18 | $C_{4}\wr C_{2}$ |
$B_{1}(2.Ru)$ | 20 | 5 | 18 | $C_{4}\wr C_{2}$ |
$B_{0}(Co_{3})$ | 20 | 6 | 18 | $C_{8}\rtimes S_{3}$ |
$B_{0}(Co_{2})$ | 20 | 7 | 16 | $U_{2}(3)$ |
$B_{0}(Th)$ | 20 | 7 | 20 | $U_{2}(3)$ |
Table 3: Morita equivalence classes of blocks with defect group
$P=5_{+}^{1+2}$: blocks of groups with non-normal defect groups

###### Remarks 7.3

(i) The two nonprincipal blocks of maximal defect of $3.SU_{3}(5)$ are
conjugate, and so Morita equivalent, in $3.SU_{3}(5).2$.

(ii) The two nonprincipal blocks of maximal defect of $3.SU_{3}(5).3$ are
conjugate, and so Morita equivalent, in $3.SU_{3}(5).6$.

(iii) The two nonprincipal blocks of maximal defect of $3.McL$ are conjugate,
and so Morita equivalent, in $3.McL.2$.

##  8 The Alperin-McKay conjecture

In this section we prove Corollary 1.3. In Table 4 we give data on the blocks
in Theorem 1.1(f-h). Inertial quotients were computed using [13], [38] and
GAP, searching within the holomorph of $p_{+}^{1+2}$ for appropriate subgroups
where data in [13] and [38] was insufficient.

$p$ | $B$ | $k_{0}(B)$ | $k_{1}(B)$ | Inertial quotient | Notes
---|---|---|---|---|---
7 | $B_{0}(He)$ | 20 | 3 | $S_{3}\times C_{3}$ |
7 | $B_{0}(He.2)$ | 25 | 6 | $S_{3}\times C_{6}$ |
7 | $B_{0}(O^{\prime}N)$ | 20 | 4 | $C_{3}\times D_{8}$ |
7 | $B_{0}(3.O^{\prime}N)$ | 20 | 4 | $C_{3}\times D_{8}$ | Remark 8.1(i)
7 | $B_{0}(O^{\prime}N.2)$ | 25 | 8 | $C_{3}\times D_{16}$ |
7 | $B_{0}(Fi_{24}^{\prime})$ | 25 | 6 | $S_{3}\times C_{6}$ |
7 | $B_{1}(3.Fi_{24}^{\prime})$ | 25 | 6 | $S_{3}\times C_{6}$ | Remark 8.1(ii)
7 | $B_{0}(Fi_{24}^{\prime}.2)$ | 35 | 6 | $C_{3}\times(C_{3}\rtimes D_{8})$ |
11 | $B_{0}(J_{4})$ | 42 | 7 | $C_{5}\times GL_{2}(3)$ |
13 | $B_{0}(M)$ | 55 | 7 | $C_{3}\times U_{2}(3)$ |
Table 4: Blocks of sporadic almost simple groups with defect group
$p_{+}^{1+2}$ for $p\geq 7$

###### Remarks 8.1

(i) The two nonprincipal blocks of maximal defect of $3.O^{\prime}N$ are
conjugate, and so Morita equivalent, in $3.O^{\prime}N.2$.

(ii) The two nonprincipal blocks of maximal defect of $3.Fi_{24}^{\prime}$ are
conjugate, and so Morita equivalent, in $3.Fi_{24}^{\prime}.2$.

Next we compute invariants in case (b) of Theorem 1.1.

###### Lemma 8.2

Consider $G=SL_{2}(p)C=SL_{2}(p)\rtimes C$ where $C\leq C_{p-1}$ so
$SL_{2}(p)\leq G\leq GL_{2}(p)$. Let $B$ be the unique $p$-block of $G$. Let
$D\in{\rm Syl}_{p}(G)$ and let $b$ be the unique block of $N_{G}(D)$.

Write $c=|C|$. Let $\psi:GL_{2}(p)\rightarrow{\rm Out}(SL_{2}(p))$, defined
via conjugation, so that $\psi(C)$ has order dividing $2$.

  1. $(a)$

(i) If $|\psi(C)|=1$, then ${\rm Irr}(B)$ consists of: $c$ linear characters;
$2c$ of degree $(p+1)/2$; $2c$ of degree $(p-1)/2$; $(p-3)c/2$ of degree
$p+1$; $(p-1)c/2$ of degree $(p-1)$; $c$ of degree $p$. Hence
$k_{0}(B)=(p+4)c+(p-1)/c$ and $k_{1}(B)=c$.

(ii) If $|\psi(C)|=2$, then ${\rm Irr}(B)$ consists of: $c$ linear characters;
$(p-2)c/2$ of degree $p+1$; $pc/2$ of degree $p-1$; $c$ of degree $p$. Hence
$k_{0}(B)=(p+1)c+(p-1)/c$ and $k_{1}(B)=c$.

In either case $l(B)=(p-1)c$.

  2. $(b)$

(i) If $|\psi(C)|=1$, then ${\rm Irr}(b)$ consists of: $(p-1)c$ linear
characters; $4c$ of degree $(p-1)/2$; $c$ of degree $p-1$; $(p-1)/c$ of degree
$(p-1)c$; $c$ of degree $p(p-1)$. Hence $k_{0}(b)=(p+4)c+(p-1)/c$ and
$k_{1}(B)=c$.

(ii) If $|\psi(C)|=2$, then ${\rm Irr}(b)$ consists of: $(p-1)c$ linear
characters; $2c$ of degree $p-1$; $(p-1)/c$ of degree $(p-1)c$; $c$ of degree
$p(p-1)$. Hence $k_{0}(b)=(p+1)c+(p-1)/c$ and $k_{1}(B)=c$.

In either case $l(b)=(p-1)c$.

Proof of Corollary 1.3. By Theorem 1.1 $B$ is Morita equivalent to one of the
blocks listed with the same inertial quotient and Külshammer-Puig class, and
the same is true of the Brauer correspondent block $b$ of $N_{G}(D)$.
Therefore it suffices to check the conjecture for the listed blocks. For those
of type (a) the result is immediate, and for type (b) it follows from Lemma
8.2. For types (c) and (d), the result follows from [33], noting that the
equivariant bijections constructed in these cases are indeed between $B$ and
$b$. Since ${\rm Out}(PSL_{3}(p))$ and ${\rm Out}(PSU_{3}(p))$ both have
cyclic Sylow $l$-subgroups for all prime $l$ (so that irreducible characters
of the quasisimple group extend to their stabilizers), the equivariant
bijection implies immediately that the conjecture holds in this case. For type
(e) the conjecture follows by comparing Tables 3 and 2. For the remaining
cases we refer to Table 4. To determine $k_{0}(b)$ and verify that it is equal
to $k_{0}(B)$, as in Section 7 we used GAP to form the holomorph of
$p_{+}^{1+2}$ and found $D\rtimes E$ through a search of its subgroups. We
computed the inertia subgroups of the nontrivial irreducible characters of $D$
in $E$, which in all cases were groups with trivial Schur multiplier. From
this we could deduce $k_{0}(b)$. $\Box$

Acknowledgments

We thank Shigeo Koshitani for some useful discussions and for drawing our
attention to [35].

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