# Elementary abelian subgroups: from algebraic groups to finite groups

Jianbei An  J. An: Department of Mathematics, University of Auckland,
Auckland, New Zealand  ,  Heiko Dietrich  H. Dietrich: School of Mathematical
Sciences, Monash University, Clayton VIC 3800, Australia  and  Alastair
Litterick  A. J. Litterick: Department of Mathematical Sciences, University of
Essex, Wivenhoe Park, Colchester, Essex CO4 3SQ, England<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We describe a new approach for classifying conjugacy classes of elementary
abelian subgroups in simple algebraic groups over an algebraically closed
field, and understanding the normaliser and centraliser structure of these.
For toral subgroups, we give an effective classification algorithm. For non-
toral elementary abelian subgroups, we focus on algebraic groups of
exceptional type with a view to future applications, and in this case we
provide tables explicitly describing the subgroups and their local structure.
We then describe how to transfer results to the corresponding finite groups of
Lie type using the Lang-Steinberg Theorem; this will be used in forthcoming
work to complete the classification of elementary abelian $p$-subgroups for
torsion primes $p$ in finite groups of exceptional Lie type. Such
classification results are important for determining the maximal $p$-local
subgroups and $p$-radical subgroups, both of which play a crucial role in
modular representation theory.

###### Key words and phrases:

algebraic groups, elementary abelian subgroups, $p$-local structure

The first author was supported by the Marsden Fund (of New Zealand), via award
numbers UOA 1626 and UOA 2030. The second author thanks Jianbei An and Eamonn
O’Brien for the hospitality during various invited research visits to the
University of Auckland. The third author acknowledges support from a Humboldt
Fellowship for Postdoctoral Researchers from the Alexander von Humboldt
Foundation, Germany. The second and third author 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/R014604/1. The authors thank Meizheng Fu for a careful reading of
the paper.

##  1\. Introduction

Many open conjectures in representation theory, like the McKay, Dade, and
Alperin Weight conjectures, have reductions to finite quasi-simple groups. For
example, Navarro & Tiep [35] have shown that the Alperin Weight Conjecture is
true for every finite group if every finite simple group satisfies a stronger
condition, namely, _AWC goodness_. (For another example, see the recent work
of Feng, Li & Zhang [19].) Verifying this latter property requires a detailed
study of the finite simple groups and their covering groups. An important role
in this study is played by so-called _$p$ -radical_ subgroups and their
$p$-local structure—that is, subgroups $R\leqslant G$ which satisfy
$R=O_{p}(N_{G}(R))$, the largest normal $p$-subgroup of the normaliser
$N_{G}(R)$—together with their normalisers $N_{G}(R)$ and centralisers
$C_{G}(R)$. The recent articles of Malle & Kessar, e.g. [27], give an
excellent survey of the current state of these conjectures and the role of
$p$-local structure. In fact, radical subgroups are relevant in many areas of
modular representation theory: for instance, defect groups of blocks are
radical, the subgroup $R$ of a weight $(R,\varphi)$ is radical, and the first
nontrivial subgroup in any radical chain is radical. If the radical subgroups
of $G$ are known, then the essential rank of the Frobenius category
$\mathcal{F}_{D}(G)$ ($D\leqslant G$ a Sylow subgroup) can be determined, cf.
[2]. Radical subgroups are also used in the study of so-called $p$-local
geometries, cf. [28]. The classification of radical subgroups therefore is an
important open problem; classifications are known for the the symmetric,
classical, and sporadic groups, as well as for some exceptional groups of Lie
type, see [6] and the references therein.

One approach to classify $p$-radical subgroups of a finite group $G$ is via
its $p$-local subgroups: A subgroup $M\leqslant G$ is _$p$ -local_ if it is
the normaliser of a nontrivial $p$-subgroup of $G$. It is _maximal $p$-local_
if $M$ is maximal with respect to inclusion among all $p$-local subgroups of
$G$. It is _local maximal_ if it is $p$-local for some prime $p$ and maximal
among all subgroups of $G$. If $G$ has a normal $p$-subgroup, then the only
maximal $p$-local subgroup is $G$ itself. Following the notation of previous
works, we say $M<G$ is _maximal-proper $p$-local_ if $M$ is $p$-local and
maximal with respect to inclusion among all proper subgroups of $G$ that are
$p$-local. Thus if $O_{p}(G)=1$ then the maximal-proper $p$-local subgroups
are exactly the maximal $p$-local subgroups. Now, if $R\leqslant G$ is
$p$-radical with $O_{p}(G)\neq R$, then $N_{G}(R)$ is $p$-local and
$N_{G}(R)\leqslant N_{G}(C)$ for every characteristic subgroup $C\leqslant R$.
In particular, $N_{G}(R)$ is contained in some maximal-proper $p$-local
$M\leqslant G$, so that $N_{G}(R)=N_{M}(R)$ and $R$ is $p$-radical in $M$.
This shows that every radical $p$-subgroup $R$ of $G$ with $O_{p}(G)\neq R$ is
radical in some maximal-proper $p$-local subgroup $M=N_{G}(R)$ of $G$. Since
$N_{G}(R)\leqslant N_{G}(\Omega_{1}(Z(R)))$, one can show that every maximal-
proper $p$-local subgroup can be realised as the normaliser of an elementary
abelian $p$-subgroup. This approach has been used successfully in recent work
[3, 9, 10, 6, 4, 5], leading to various classifications of $p$-local and
$p$-radical subgroups in finite exceptional groups of Lie type. Similar to
$p$-radical groups, $p$-local groups also play an important role in group
theory. For example, large parts of the classification of the finite simple
groups are based on the analysis of $p$-local subgroups; one reason for this
is that the fusion of $p$-elements is controlled by normalisers of
$p$-subgroups, by the Alperin-(Goldschmidt) Fusion Theorem. We note that Cohen
et al. [16] classified local maximal subgroups of exceptional groups of Lie
type. However, not every maximal-proper $p$-local subgroup is local maximal,
and the details obtained in the classification of maximal-proper $p$-local
subgroups have been proved to be useful for the classification of radical
subgroups.

All of this highlights the importance of knowing the elementary abelian
$p$-subgroups of a finite group of Lie type; Griess [22, Section 1] lists more
references supporting this. With the knowledge of the elementary abelian
subgroups, one can attempt a classification of the maximal-proper $p$-local
and $p$-radical subgroups. Recent efforts in this direction have focused
directly on the finite groups involved (see [3, 9, 10, 6, 4] for groups of
type $F_{4}$, $E_{6}$, $E_{7}$). In contrast, this paper takes an alternative
approach beginning with an ambient algebraic group over an algebraically
closed field. In this case, the rich geometric and Lie-theoretic structure of
the group drastically simplifies many arguments, and indeed there are a number
of existing results in this case [22, 40, 24, 11] which we will build upon
here. Supposing one has complete subgroup structure information for a given
algebraic group in positive characteristic, the Lang-Steinberg Theorem (cf.
[32, §21.2]) then gives a powerful technique for transferring results to the
corresponding finite groups of Lie type. This both streamlines arguments, and
avoids some of the lengthy ad-hoc calculations required in previous papers.

We note that the results of this paper have already been used successfully in
[5, 6] to classify the maximal $3$-local and $3$-radical subgroups in the
finite groups of type $E_{6}$. It is also used in [8] for work on Donovan’s
conjecture for blocks with extra-special defect groups $p_{+}^{1+2}$, in [1]
to classify weighted subgroups of quasi-isolated $2$-blocks of $F_{4}(q)$, and
in the (ongoing) PhD thesis of Fu to classify elementary abelian subgroups in
finite classical groups. In forthcoming work, we apply the results of this
paper to classify the elementary abelian $p$-subgroups in the finite
exceptional groups of Lie type for small (torsion) primes. In particular, our
results are also a significant step toward a classification of the maximal-
proper $p$-local subgroups in the finite groups of Lie type; this research
will be published in a separate paper.

###  1.1. Main results and structure of the paper

Let $p$ be a prime and let $G$ be a simple algebraic group over an
algebraically closed field of characteristic $\ell\geqslant 0$. If $\ell>0$,
then we choose a Steinberg endomorphism $F$ of $G$, with corresponding fixed-
point subgroup $G^{F}$, a finite group of Lie type. If $\ell=p$ then the
$p$-radical subgroups on $G^{F}$ are known by the Borel–Tits Theorem [21,
Corollary 3.1.5], hence we assume throughout that $\ell\neq p$. Thus each
elementary abelian $p$-subgroup of $G$ consists of semisimple elements, and
therefore normalises a conjugate of a fixed maximal torus $T\leqslant G$. Our
strategy is to consider separately the case of _toral_ subgroups (those with a
conjugate contained in $T$) and _non-toral_ subgroups (those which are not
toral). While we consider the toral elementary subgroups for all simple
algebraic groups, we restrict ourselves to exceptional groups for the non-
toral ones; classification of non-toral elementary abelian groups in the
classical case is more difficult (and is sometimes related to the
classification of certain codes, see [22, Table 1]). Our main results are the
following:

  * (1)

A constructive method to classify the toral elementary abelian $p$-subgroups
(up to conjugacy) in a simple algebraic group $G$ and to determine information
on their local structure, see Section 5. Our method translates to a practical
algorithm that we have implemented for the computer algebra system Magma [13];
our implementation is available under the link provided in [7].

  * (2)

A complete classification of the non-toral elementary abelian $p$-subgroups
(up to conjugacy) in an exceptional simple algebraic group $G$ and their local
structure, see Propositions 6.3 and 6.3 in Section 6.

  * (3)

A description of how (1) and (2) can be used to classify the elementary
abelian $p$-subgroups (up to conjugacy) and their local structure in a finite
simple group $G^{F}$, see Section 4.

######  Remark 1.1.

In fact, the method for classifying subgroups in (1) and (2) works for any
subgroup of order coprime to the characteristic $\ell$, and the arguments
relating to local structure extend too when the subgroups are also abelian.
However, our results on torality and local structure depend implicitly on the
fact that the subgroup in question is an abelian $p$-group; for this reason
and the reasons above, we keep our focus on the elementary abelian case.

First consider toral subgroups. For a maximal torus $T$ of $G$, it is well
known that the Weyl group $N_{G}(T)/T$ controls conjugacy and determines the
normaliser structure of subgroups of $T$. By viewing $G$ as the group of
points of a suitable group scheme, it follows that the classification of toral
elementary abelian $p$-subgroups of $G$ is independent of the characteristic
$\ell$ as long as $\ell\neq p$, which we assume throughout. Importantly, this
shows that all of these calculations can be performed in a suitable finite
group of Lie type, which gives rise to a practical computational approach
toward a classification. We give full details in Section 5.

Turning to non-toral elementary abelian subgroups in Section 6, the number of
classes of such subgroups is bounded by an absolute constant, and in many
cases these have been described elsewhere in the literature: For example, the
maximal non-toral elementary abelian subgroups in complex groups are described
by Griess [22]. Complete information on non-toral subgroups for $p\neq 2$ in
groups over $\mathbb{C}$ is given by Andersen et al. [11]. For $p=2$, much
information is provided by Yu [40] for adjoint compact groups. Again, all
these results carry over into any characteristic different from $p$. We
present explicit tables (including new information on the local structure) and
we use the opportunity to correct some typographic errors in the ancillary
data of [40].

Assuming $\ell>0$ and knowledge of the elementary abelian subgroups and their
local structure in $G$, in Section 4 we use a consequence of the Lang-
Steinberg Theorem to see how the $G$-classes of subgroups split into subgroups
of the finite groups $G^{F}$. If some $G$-conjugate of an elementary abelian
subgroup $E$ lies in $G^{F}$, then the $G^{F}$-subgroup classes arising from
$E$ correspond to $F$-classes in $N_{G}(E)/C_{G}(E)^{\circ}$ of elements
contained in $C_{G}(E)/C_{G}(E)^{\circ}$, and these are known by our previous
calculations. The normaliser and centraliser structure also follow in short
order.

In Appendices A and B we give various ancillary data which we computed (or re-
computed) in the course of proving our results for non-toral subgroups, in
particular, for $p=2$ and types $E_{7}$ and $E_{8}$.

##  2\. Notation

Throughout, unless stated otherwise, $G$ is a simple algebraic group, defined
over an algebraically closed field $K$ of characteristic $\ell\geqslant 0$.
When discussing elementary abelian $p$-groups, we always assume $p\neq\ell$,
restating this assumption when necessary. We treat $G$ as a Zariski-closed
subgroup of some general linear group over the algebraically closed field,
except in Section 3.1 where we briefly mention the scheme-theoretic background
necessary to transfer results between groups over fields of different
characteristics. We add subscripts “sc” and “ad” to denote the simply-
connected and adjoint group, respectively, of a given type.

For $q$ a prime power, we denote by $\mathbb{F}_{q}$ the finite field with $q$
elements. We denote by $T$ a fixed maximal torus of $G$, and define the Weyl
group of $G$ as $W=N_{G}(T)/T$; this does not depend on the choice of $T$
since all maximal tori in a linear algebraic group are conjugate [32,
Corollary 6.5]. Recall that $G$ has an associated root datum, consisting of a
root system $\Phi$, dual root system $\Phi^{\vee}$, character lattice
$X=\operatorname{Hom}(T,K^{\ast})\supseteq\mathbb{Z}\Phi$ and co-character
lattice $Y=\operatorname{Hom}(K^{\ast},T)\supseteq\mathbb{Z}\Phi^{\vee}$, with
a natural pairing $\left<-,-\right>\colon X\times Y\to\mathbb{Z}$. There are
natural notions of homomorphism and isomorphism of root data, and $G$ is
determined up to isomorphism by the field $K$ and its root datum, see [25,
II.1.13].

We follow the convention of Malle and Testermann [32, Section 21.2] and call
an endomorphism $F\colon G\to G$ of a linear algebraic group $G$ a _Steinberg
morphism_ if some power $F^{m}$ is a _Frobenius morphism_ with respect to some
$\mathbb{F}_{q}$-structure, that is, $F^{m}$ is induced by the $q$-power map
$K\to K$. A Steinberg morphism is an automorphism of abstract groups, but not
necessarily of algebraic groups. As usual, if $X$ is a set and $F$ is a
function $X\to X$, then we denote by $X^{F}$ the fixed-point set of $F$. If
$F$ is a Steinberg morphism of a linear algebraic group with $F^{m}$ a
Frobenius morphism, then $G^{F}\leqslant G^{F^{m}}$ are both finite groups. We
note that this implies $F$ is a Steinberg morphism as defined in Gorenstein et
al. [21, Definition 1.15.1]. Lastly, we mention a celebrated result of
Steinberg which shows that if $G$ is simple (the case of most interest here),
then every non-trivial endomorphism of $G$ is either an automorphism of
algebraic groups or a Steinberg morphism, and the latter occurs if and only if
$G^{F}$ is finite, see [32, Theorem 21.5].

For an algebraic group $H$, we denote by $H^{\circ}$ the connected component
of the identity element. If $X$ is a finite group then $O_{p}(X)$ denotes the
largest normal $p$-subgroup of $X$, and if $P$ is a $p$-group, then
$\Omega_{k}(P)$, with $k\geqslant 1$ an integer, denotes the subgroup
generated by elements of order dividing $p^{k}$.

Let $A,B$ be groups and $l,n,m$ be positive integers. Let $p$ be a prime. We
denote by $A\times B$ the direct product of $A$ and $B$. We sometimes write
$m$ for the cyclic group of size $m$, and we denote by $A^{n}$ the direct
product of $n$ copies of $A$. This leads to an ambiguity if $m=p^{n}$ is a
prime power; we avoid this by writing $p^{n}$ only for the direct product of
groups of order $p$, and $m$ for the cyclic group of order $m$. An extension,
split extension, and central extension of $A$ by $B$ are respectively denoted
$A.B$, $A{:}B$, and $A\circ B$; here $A$ is the normal subgroup. We read
$A.B.C$ as $(A.B).C$, and similarly for other products of groups. For group
elements $g,h\in A$ we denote conjugation as usual by ${}^{g}h=ghg^{-1}$ and
$h^{g}=g^{-1}hg$. The notation $p^{1+2n}$ denotes an extraspecial group; there
are two isomorphism types of these, which are denoted by $p_{+}^{1+2n}$ and
$p_{-}^{1+2n}$. The alternating and symmetric group on $n$ points are denoted
by $\operatorname{Alt}_{n}$ and $\operatorname{Sym}_{n}$, respectively. The
dihedral and generalised quaternion group of order $n$ are ${\rm Dih}_{n}$ and
$Q_{n}$. We write $\operatorname{GL}_{n}(q)$ for the general linear group of
degree $n$ over the field with $q$ elements, and $\operatorname{SL}_{n}(q)$,
$\operatorname{SU}_{n}(q)$, ${\rm Sp}_{n}(q)$, and
$\operatorname{O}^{\pm}_{n}(q)$ for the classical groups (special linear,
unitary, symplectic, and orthogonal group). Other notation is introduced at
the appropriate places.

##  3\. Preliminaries

###  3.1. Independence from the characteristic

The following result is of vital importance for establishing our main results.
In the present form this is due to Larsen [23, Appendix A], but see also [11,
p. 137], [17, Section 3], [22, Appendix A], and [20]. It tells us that the
classification of elementary abelian $p$-subgroups of $G$ is independent of
the defining characteristic $\ell$, as long as it is different from $p$; in
particular, it shows that the classification of elementary abelian
$p$-subgroups of a semisimple algebraic group $G(K)$ coincides with that of
the corresponding complex Lie group $G(\mathbb{C})$. The proof makes use of
deep algebraic geometry, interpreting $G$ as a group scheme over a ring of
Witt vectors, which is an integral domain of characteristic $0$ that maps onto
an algebraically closed field of positive characteristic.

######  Proposition 3.1 ([23, Theorem 1.22]).

Let $\ell$ be a prime number and $G$ a semisimple split group scheme; let $X$
be a finite group of order coprime to $\ell$. For any algebraically closed
fields $K_{0}$ and $K_{\ell}$ of characteristic $0$ and $\ell$, respectively,
the sets $\operatorname{Hom}(X,G(K_{0}))/G(K_{0})$ and
$\operatorname{Hom}(X,G(K_{\ell}))/G(K_{\ell})$ have the same cardinality,
where the quotients are taken with respect to the natural conjugation action
of the groups on the homomorphism sets. Furthermore, the process of reduction
modulo $\ell$ induces a bijection.

The ‘process of reduction modulo $\ell$’ here entails showing that every
finite subgroup of $G(K_{0})$ has a conjugate contained in $G(R)$, where $R$
is a suitable ring of Witt vectors that maps onto an algebraically closed
subfield of $K_{\ell}$, and then considering the induced map $G(R)\to
G(K_{\ell})$; we refer to [23, Appendix A] and [20] for more details.

Recall, for instance from [11, Proposition 8.4], that a complex algebraic
group $G=G(\mathbb{C})$ has a maximal compact subgroup in the Hausdorff
topology, say $H$, which is unique up to conjugacy and _strongly controls
fusion_ : if $U^{g}=V$ for subsets $U,V\subseteq H$ and $g\in G$, then there
is some $h\in H$ such that $u^{g}=u^{h}$ for all $u\in U$. In particular, if
$U^{g}=U$, then $g\in C_{G}(U)N_{H}(U)$, so $N_{G}(U)=C_{G}(U)N_{H}(U)$.
Moreover, if $U\leqslant G$ is a finite subgroup, then $N_{G}(U)/C_{G}(U)$
embeds into the (finite) automorphism group $\operatorname{Aut}(U)$, and since
$C_{G}(U)^{\circ}$ has finite index in $C_{G}(U)$, it has finite index in
$N_{G}(U)$; thus, we have $N_{G}(U)^{\circ}=C_{G}(U)^{\circ}$, and therefore
$N_{G}(U)/N_{G}(U)^{\circ}=N_{H}(U)C_{G}(U)/C_{G}(U)^{\circ}$. Based on all
this, the next lemma shows that we can view the $p$-local structure of $G$
inside $H$.

######  Lemma 3.2.

Let $G$ be a reductive complex algebraic group with a maximal compact subgroup
$H$, and let $E\leqslant H$ be elementary abelian. Then the following hold:

  * (a)

The inclusion $H\hookrightarrow G$ induces a bijection on conjugacy classes of
elementary abelian subgroups of $G$.

  * (b)

The group $C_{G}(E)^{\circ}\cap H$ is a maximal compact subgroup of
$C_{G}(E)^{\circ}$, with the same root datum.

  * (c)

$N_{H}(E)=(H\cap N_{G}(E)^{\circ}).(N_{G}(E)/N_{G}(E)^{\circ})$ and
$C_{H}(E)=(H\cap C_{G}(E)^{\circ}).(C_{G}(E)/C_{G}(E)^{\circ})$.

###### Proof.

By uniqueness of $H$, each compact subgroup of $G$ has a $G$-conjugate
contained in $H$. Uniqueness of this $G$-conjugate up to $H$-conjugacy follows
since $H$ strongly controls fusion in $G$; this proves a). Next, by [11,
Proposition 8.4(3)] the group $C_{H}(E)=H\cap C_{G}(E)$ is a maximal compact
subgroup of $C_{G}(E)$; this is reductive, and the inclusion $C_{H}(E)\to
C_{G}(E)$ sends a maximal torus of $C_{H}(E)$ into one of $C_{G}(E)^{\circ}$,
matching up the corresponding root data, giving b). For c), note that each
coset of $N_{G}(E)^{\circ}$ in $N_{G}(E)$ contains a translate of
$N_{G}(E)^{\circ}\cap H$, which is a maximal compact subgroup of
$N_{G}(E)^{\circ}=C_{G}(E)^{\circ}$. The union of these translates is a finite
extension of a compact group, hence is compact, and therefore $N_{H}(E)$ meets
each such coset. Identical reasoning applies for the centraliser of $E$. (A
more detailed justification for parts (b) and (c) can be obtained using the
complexification functor from compact Lie groups to complex reductive groups,
as detailed e.g. in [26, Theorems 1, 2].) ∎

Thus the classification of elementary abelian subgroups and their local
structure in a complex reductive group can be viewed inside compact real Lie
groups. We will refer in particular to [11, 22, 40].

Proposition 3.1 can already be applied to finite subgroups of $N_{G}(E)$ and
$C_{G}(E)$ for an elementary abelian subgroup $E\leqslant G$, but stronger
results are available since $N_{G}(E)$ and $C_{G}(E)$ can themselves be viewed
as the points of a group scheme. The following lemma summarises a series of
results of Friedlander & Mislin [20] which make use of this viewpoint. We note
that the results quoted from [20] require an elementary abelian $p$-subgroup
with $p\neq\ell$, but their proofs should also hold more generally.

######  Lemma 3.3.

Let $K_{0}$ and $K_{\ell}$ be algebraically closed fields of characteristic
$0$ and $\ell$, respectively, and let $p\neq\ell$ be a prime. Let $E$ and
$E^{\prime}$ respectively be elementary abelian $p$-subgroups of $G(K_{0})$
and $G(K_{\ell})$ which correspond under the bijection of Proposition 3.1.
Then reduction-mod-$\ell$ allows us to identify the root data of the reductive
groups $C_{G(K_{0})}(E)^{\circ}$ and $C_{G(K_{\ell})}(E^{\prime})^{\circ}$,
and also induces isomorphisms

$\displaystyle N_{G(K_{0})}(E)/C_{G(K_{0})}(E)\phantom{{{}^{\circ}}}$
$\displaystyle\cong
N_{G(K_{\ell})}(E^{\prime})/C_{G(K_{\ell})}(E^{\prime})\phantom{{}^{\circ}},$
$\displaystyle N_{G(K_{0})}(E)/C_{G(K_{0})}(E)^{\circ}$ $\displaystyle\cong
N_{G(K_{\ell})}(E^{\prime})/C_{G(K_{\ell})}(E^{\prime})^{\circ},$
$\displaystyle C_{G(K_{0})}(E)/C_{G(K_{0})}(E)^{\circ}$ $\displaystyle\cong
C_{G(K_{\ell})}(E^{\prime})/C_{G(K_{\ell})}(E^{\prime})^{\circ}.$

Moreover, the order of the component group
$C_{G(K_{0})}(E)/C_{G(K_{0})}(E)^{\circ}$ is a power of $p$.

###### Proof.

We first claim that without loss of generality we can take $K_{0}=\mathbb{C}$
and $K_{\ell}=\overline{\mathbb{F}}_{\ell}$, the algebraic closure of the
field $\mathbb{F}_{\ell}$ of $\ell$ elements. Since $\ell\neq p$, all
representations of a finite elementary abelian $p$-group in characteristic $0$
and characteristic $\ell$ are semisimple, see Maschke’s Theorem. This shows
that $E$ and $E^{\prime}$ are _linearly reductive_ in the sense of [12, §2],
thus $E$ and $E^{\prime}$ are _$G$ -completely reducible_ [12, Lemma 2.6], and
our first claim is then [12, Corollary 5.5].

Now we interpret $E$ as the $\mathbb{C}$-points of a finite subgroup scheme of
$G$, defined over an appropriate ring of Witt vectors $R$, where $R$ comes
equipped with a fixed embedding into $\mathbb{C}$ and a surjection onto
$\overline{\mathbb{F}}_{\ell}$. By hypothesis, the image of $E$ in
$G(\overline{\mathbb{F}}_{\ell})$ is conjugate to $E^{\prime}$, so we may
replace $E^{\prime}$ by this image without loss of generality. Then
$N_{G(\mathbb{C})}(E)$ and $N_{G(\overline{\mathbb{F}}_{\ell})}(E^{\prime})$
are the $\mathbb{C}$-points and $\overline{\mathbb{F}}_{\ell}$-points of the
scheme-theoretic normaliser of $E$ in $G$, which we denote $N_{G}(E)$, and
similarly $C_{G(\mathbb{C})}(E)$ and
$C_{G(\overline{\mathbb{F}}_{\ell})}(E^{\prime})$ are the points of the
scheme-theoretic centraliser $C_{G}(E)$. Then [20, Corollary 4.3 and Theorem
4.4] tell us that $N_{G}(E)$ and $C_{G}(E)$ are _generalised reductive groups_
over $R$ (see [20, Definition 2.1]), and applying [20, Proposition 3.1(i)] to
each of these, we deduce that the quotient by the connected component is the
same over $\mathbb{C}$ and $\overline{\mathbb{F}}_{\ell}$. This gives the
second and third isomorphisms above. The first isomorphism is derived in the
proof of [20, Theorem 4.4].

Lastly, since the scheme-theoretic centraliser $C_{G}(E)$ is a generalised
reductive group, by definition the quotient of $C_{G}(E)$ by its identity
component is a finite _étale group scheme_ which by [20, Corollary 4.3] has
order coprime to $\ell$; note that in [20] the quotient by the identity
component is denoted $\pi_{0}(-)$. We can repeat this argument for any
$\ell\neq p$, replacing $\ell$ and $R$ as appropriate, and so the order is
coprime to every prime except possibly $p$; in other words, its order is a
power of $p$. ∎

###  3.2. Torality

As hinted at in the introduction, the behaviour of elementary abelian
subgroups admits a stark dichotomy depending on whether the subgroups in
question are toral or not. In general, toral subgroups are much more well-
behaved, and non-toral subgroups are comparatively rare, as illustrated by the
following result of Steinberg. To state it, recall that a prime $p$ is called
a _torsion prime_ for a reductive group $G$ if some subsystem subgroup of $G$
has $p$-torsion in its fundamental group [39, Definition 2.1]. Explicitly, if
$G$ is simple, then a torsion prime satisfies: $p\mid n+1$ for $G$ of type
$A_{n}$; $p=2$ for $G$ of type $B_{n}$, $C_{n}$, $D_{n}$, $G_{2}$; $p\leqslant
3$ for $G$ of type $F_{4}$, $E_{6}$, $E_{7}$; $p\leqslant 5$ for $G$ of type
$E_{8}$; the exact possibilities depend on the isogeny type of the group and
are listed in [39, Lemma 2.5] and [32, Table 9.2]. If $G$ is not simple, then
$p$ is torsion for $G$ precisely when it is torsion for some simple factor.

######  Theorem 3.4 ([39, Theorem 2.28]).

Let $G$ be a reductive algebraic group in characteristic $\ell$ and let
$p\neq\ell$ be a prime. The following conditions are equivalent:

  * (a)

The prime $p$ is not a torsion prime for $G$.

  * (b)

The centraliser $C_{G}(E)$ is connected for every elementary abelian
$p$-subgroup $E$.

  * (c)

Every elementary abelian $p$-subgroup of $G$ is toral.

We close this subsection with the following useful lemma.

######  Lemma 3.5.

Let $G$ be a reductive algebraic group in characteristic $\ell$ and let
$E\leqslant G$ be an elementary abelian $p$-subgroup, where $p\neq\ell$.

  * (a)

The subgroup $E$ is toral in $G$ if and only if $E\leqslant C_{G}(E)^{\circ}$.

  * (b)

Let $Z\leqslant G$ be central. Then $E$ is toral in $G$ if and only if $EZ/Z$
is toral in $G/Z$.

  * (c)

Suppose that $G$ is complex and view the Lie algebra of $G$ as a
$\mathbb{C}E$-module with character $\chi_{L}$. Then

$\dim C_{G}(E)=\frac{1}{|E|}\sum\nolimits_{x\in E}\chi_{L}(x).$

###### Proof.

Suppose $E$ is contained in some maximal torus $T$ of $G$. Since $T$ is
connected and abelian, we have $T\leqslant C_{G}(E)^{\circ}$ and so
$E\leqslant C_{G}(E)^{\circ}$. Conversely, suppose $E\leqslant
C_{G}(E)^{\circ}$ and recall that $C_{G}(E)^{\circ}$ is reductive, see [32,
Theorem 14.2]. Since the centre of a connected reductive group is contained in
every maximal torus, $E$ is contained in every maximal torus of
$C_{G}(E)^{\circ}$, in particular $E$ is contained in some torus of $G$. This
proves a). Parts b) and c) are proved in [11, Theorem 8.2]. ∎

###  3.3. Semisimple elements of small order

In our discussion of non-toral elementary abelian subgroups we will require
the information provided in Table 1 below, on certain conjugacy classes of
semisimple elements in exceptional groups $G$. We give the centraliser of each
such element $g$, as well as the trace (Brauer character) $\chi_{L}(g)$ on the
Lie algebra $L(G)$ of $G$; for $G\neq E_{8}(\mathbb{C})$ we also give the
trace $\chi_{\rm min}(g)$ on a non-trivial module of least dimension. By the
results of Section 3.1, this information remains the same for the
corresponding algebraic group in any characteristic coprime to the element
order. These results can be obtained using the algorithm of Moody and Patera
[34], but they have already appeared throughout the literature, in particular
[18, Table 2], [15, Tables 4, 6], [34, Table 10], and [21, Tables 4.3.1,
4.3.2, and 4.7.1]. We have chosen our class labels to be consistent with these
references.

If $r$ is a prime, then $r^{k}\text{X}$ stands for a conjugacy class of
elements of order $r^{k}$. The notation $3{\rm B}[2]$ and $3{\rm E}[2]$
indicates that the squares of these elements form distinct conjugacy classes,
and $4{\rm A}[{\rm A}]$ and $4{\rm H}[{\rm A}]$ indicate that the square of
these elements lies in class $2{\rm A}$. A class distribution
$r\text{X}_{a}\text{Y}_{b}\text{Z}_{c}$ for a subgroup $E$ indicates that
there are $a$, $b$, $c$ elements in $r\text{X}\cap E$, $r\text{Y}\cap E$, and
$r\text{Z}\cap E$, respectively. In the description of the centraliser,
$T_{j}\cong(\mathbb{C}^{\times})^{j}$ is a $j$-dimensional torus. The element
$\omega_{3}$ is a fixed cube root of of unity. The isogeny types of the groups
in the centralisers can be determined from the action of the groups on $L(G)$
and the minimal module, listed for instance in [29, §8]: the correct group is
the smallest group (in the sense of taking quotients) which acts faithfully on
some composition factor of $L(G)$ if $G$ is adjoint, or of the minimal module
if $G$ is simply connected of type $E_{6}$ or $E_{7}$.

Table 1. Certain elements of small order in simply connected algebraic groups

$\boldsymbol{G}$ Class $\boldsymbol{C_{G}(x)}$ $\boldsymbol{\chi_{L}}$
$\boldsymbol{\chi_{\rm min}}$ $G_{2}(\mathbb{C})$ 2A
$\operatorname{SL}_{2}(\mathbb{C})\circ\operatorname{SL}_{2}(\mathbb{C})$ $-2$
$-1$ $F_{4}(\mathbb{C})$ 2A ${\rm
Sp}_{6}(\mathbb{C})\circ\operatorname{SL}_{2}(\mathbb{C})$ $-4$ $2$ 2B
$\operatorname{Spin}_{9}(\mathbb{C})$ $20$ $-6$ 3A ${\rm
Sp}_{6}(\mathbb{C})\circ T_{1}$ $7$ $8$ 3C
$\operatorname{SL}_{3}(\mathbb{C})\circ\operatorname{SL}_{3}(\mathbb{C})$ $-2$
$-1$ 3D $\operatorname{Spin}_{7}(\mathbb{C})\circ T_{1}$ $7$ $-1$
$E_{6,{\text{\rm sc}}}(\mathbb{C})$ 2A
$\operatorname{SL}_{2}(\mathbb{C})\circ\operatorname{SL}_{6}(\mathbb{C})$ $-2$
$3$ 2B $\operatorname{Spin}_{10}(\mathbb{C})\circ T_{1}$ $14$ $-5$ 3A
$\operatorname{SL}_{6}(\mathbb{C})\circ T_{1}$ $15$ $9$ 3B[2]
$\operatorname{SL}_{6}(\mathbb{C})\circ T_{1}$ $15$ $9\omega_{3}$ 3C
$\operatorname{SL}_{3}(\mathbb{C})\circ\operatorname{SL}_{3}(\mathbb{C})\circ\operatorname{SL}_{3}(\mathbb{C})$
$-3$ $0$ 3D $\operatorname{Spin}_{8}(\mathbb{C})\circ T_{2}$ $6$ $0$ 3E[2]
$E_{6,{\text{\rm sc}}}(\mathbb{C})$ $78$ $27\omega_{3}$ $\boldsymbol{G}$ Class
$\boldsymbol{C_{G}(x)}$ $\boldsymbol{\chi_{L}}$ $\boldsymbol{\chi_{\rm min}}$
$E_{7,{\text{\rm sc}}}(\mathbb{C})$ 2A $E_{7,{\text{\rm sc}}}(\mathbb{C})$
$133$ $-56$ 2B
$\operatorname{SL}_{2}(\mathbb{C})\circ\operatorname{Spin}_{12}(\mathbb{C})$
$5$ $8$ 2C
$\operatorname{SL}_{2}(\mathbb{C})\circ\operatorname{Spin}_{12}(\mathbb{C})$
$5$ $-8$ 3A $\operatorname{SL}_{7}(\mathbb{C})\circ T_{1}$ $7$ $-7$ 3B
$E_{6,{\text{\rm sc}}}(\mathbb{C})\circ T_{1}$ $52$ $-25$ 3C
$\operatorname{SL}_{6}(\mathbb{C})\circ\operatorname{SL}_{3}(\mathbb{C})$ $-2$
$2$ 3D
$\operatorname{SL}_{2}(\mathbb{C})\circ\operatorname{Spin}_{10}(\mathbb{C})\circ
T_{1}$ $7$ $2$ 3E $\operatorname{Spin}_{12}(\mathbb{C})\circ T_{1}$ $34$ $20$
4A[A] $\operatorname{SL}_{8}(\mathbb{C})/\left<-I\right>$ $-7$ $0$ 4H[A]
$E_{6,{\text{\rm sc}}}(\mathbb{C})\circ T_{1}$ $25$ $0$ $E_{8}(\mathbb{C})$ 2A
$\operatorname{SL}_{2}(\mathbb{C})\circ E_{7,{\text{\rm sc}}}(\mathbb{C})$
$24$ $-$ 2B $\operatorname{HSpin}_{16}(\mathbb{C})$ $-8$ $-$ 3A
$\operatorname{SL}_{9}(\mathbb{C})/\left<\omega_{3}I\right>$ $-4$ $-$ 3B
$\operatorname{SL}_{3}(\mathbb{C})\circ E_{6,{\text{\rm sc}}}(\mathbb{C})$ $5$
$-$ 3C $\operatorname{Spin}_{14}(\mathbb{C})\circ T_{1}$ $14$ $-$ 3D
$E_{7,{\text{\rm sc}}}(\mathbb{C})\circ T_{1}$ $77$ $-$ 5C
$\operatorname{SL}_{5}(\mathbb{C})\circ\operatorname{SL}_{5}(\mathbb{C})$ $-2$
$-$

We will also refer to the inclusions of classes in Table 2, induced by the
inclusions of simply connected groups $F_{4}(\mathbb{C})<E_{6,{\text{\rm
sc}}}(\mathbb{C})<E_{7,{\text{\rm sc}}}(\mathbb{C})<E_{8}(\mathbb{C})$. Some
of this information is given in the above references, but it can also be
determined directly from the character values and the known action of each
group on the Lie algebra and minimal module of the next.

Table 2. Inclusions of certain conjugacy classes $F_{4}$ class |  |  $E_{6}$ class |  |  $E_{7}$ class |  |  $E_{8}$ class
---|---|---|---|---|---|---
2A | $\to$ | 2A | $\to$ | 2B | $\to$ | 2A
2B | $\to$ | 2B | $\to$ | 2C | $\to$ | 2B
|  |  |  | 2A | $\to$ | 2A

##  4\. From algebraic groups to finite groups

Let $G$ be a connected reductive algebraic group over the algebraically closed
field $K$ of characteristic $\ell>0$, with maximal torus $T$ and Weyl group
$W$. In the following we suppose that a Steinberg endomorphism $F$ of $G$ is
given, that $T$ is $F$-stable, and that we know, up to $G$-conjugacy, the
elementary abelian subgroups of $G$ with some representative in $G^{F}$, as
well as their local structure in $G$. Given this information, we show how to
derive a classification of the elementary abelian subgroups of the finite
group $G^{F}$, together with their local structure. The main tool for
establishing this result is an application of the Lang-Steinberg Theorem. Note
that the results of this section hold for all finite subgroups; we will return
to the particular case of elementary abelian $p$-subgroups in subsequent
sections when applying these.

###  4.1. The correspondence

If $A\leqslant G^{F}$, then the _$F$ -class_ of a coset $uN_{G}(A)^{\circ}$ in
$N_{G}(A)/N_{G}(A)^{\circ}$ is the subset $\\{F(g)ug^{-1}N_{G}(A)^{\circ}:g\in
N_{G}(A)\\}$. Moreover, recall that $N_{G}(A)^{\circ}=C_{G}(A)^{\circ}$ if
$A\leqslant G$ is finite.

######  Proposition 4.1.

Let $A\leqslant G$ be a finite subgroup such that $F(A)$ and $A$ are conjugate
in $G$. The following hold:

  * (a)

There exists a $G$-conjugate of $A$ which is $F$-stable.

  * (b)

Suppose $A$ has a $G$-conjugate in $G^{F}$. Replacing $A$ by this conjugate,
there is a bijection between $G^{F}$-classes of subgroups of $G^{F}$ which are
$G$-conjugate to $A$, and $F$-classes in $N_{G}(A)/C_{G}(A)^{\circ}$ contained
in $C_{G}(A)/C_{G}(A)^{\circ}$: the $F$-class of $w\in
C_{G}(A)/C_{G}(A)^{\circ}$ corresponds to the $G^{F}$-class of subgroups with
representative $A_{w}={{}^{g}}A$, where $g\in G$ is chosen with
$g^{-1}F(g)C_{G}(A)^{\circ}=w$.

###### Proof.

To begin, $G$ clearly acts on $M=\\{{}^{g}A:g\in G\\}$ transitively. Also $F$
acts on $M$ since $F(A)$ is $G$-conjugate to $A$, and this action is
compatible with the $G$-action since $F({}^{g}A)={}^{F(g)}F(A)$ for $g\in G$.
Note that $A$ is closed since it is finite, hence so are $N_{G}(A)$ and
$C_{G}(A)$. Lang-Steinberg Theorem then implies the existence of an $F$-stable
conjugate of $A$, see [32, Theorem 21.11], so (a) holds, and replacing $A$ by
such $F$-stable conjugate there is a bijection between

$\\{G^{F}\text{-orbits on }M^{F}\\}\quad\text{and}\quad\\{F\text{-classes in
}N_{G}(A)/N_{G}(A)^{\circ}\\}.$

Now suppose $A$ is fixed point-wise by $F$, so $A\leqslant G^{F}$, and
abbreviate $W(A)=N_{G}(A)/C_{G}(A)^{\circ}$. As shown in the proof of [32,
Theorem 21.11], the $F$-class of $w\in W(A)$ corresponds to the $G^{F}$-class
of subgroups with representative $A_{w}={{}^{g}}A$, where $g\in G$ satisfies
$g^{-1}F(g)C_{G}(A)^{\circ}=w$. For such an element $g$, and for any $a\in A$,
we have $F(gag^{-1})=F(g)aF(g)^{-1}=g(\dot{w}a\dot{w}^{-1})g^{-1}$, for some
lift $\dot{w}$ of $w$ to $N_{G}(A)$. Therefore $F$ acts on $A_{w}$ as the map
$a\mapsto\dot{w}a\dot{w}^{-1}$ acts on $A$. In particular, $A_{w}^{F}=A_{w}$
if and only if $w\in C_{G}(A)/C_{G}(A)^{\circ}$, so that the $F$-class of $w$
consists of elements in $C_{G}(A)/C_{G}(A)^{\circ}$; thus (b) holds. ∎

We note that Proposition 4.1 works for both toral and non-toral elementary
abelian subgroups, and for both twisted and untwisted Steinberg morphisms.
Importantly, Proposition 4.1 allows us to determine all $G^{F}$-classes of
elementary abelian subgroups if we know the $G$-classes of these subgroups in
$G$ with representatives in $G^{F}$ (whenever they exist). One remaining
problem is to find the latter, and Corollary 4.2 below will be useful for
that. Clearly, if $A\leqslant G$ and there is $g\in G$ such that
$B={}^{g}A\leqslant G^{F}$, then $F(B)=B$ and so $A$ and $F(A)$ are conjugate.
However, the converse is not always true. For instance, let
$G=\operatorname{GL}_{n}(K)$ where $K$ has characteristic $2$, let $T$ be the
subgroup of diagonal matrices, and let $F$ be a Steinberg morphism inducing
$x\mapsto x^{2}$ on $T$. Then $T^{F}=1$ and $G^{F}$ has no elementary abelian
$p$-subgroups of rank $n$ for $p\neq 2$, even though $T$ contains an
$F$-stable elementary abelian $p$-subgroup of rank $n$; the action of $F$ on
$T_{(p)}=\Omega_{1}(O_{p}(T))$ is then non-trivial. Despite this subtlety, it
remains true that for any given finite subgroup of $G$, there is some $F$ such
that $G^{F}$ contains a conjugate of this subgroup, in which case the results
of this section tell us exactly how many such classes arise, as well as their
normaliser structure.

######  Corollary 4.2.

Let $A\leqslant G$ be a finite subgroup.

  * (a)

There is a $G$-conjugate of $A$ in $G^{F}$ if and only if $A$ and $F(A)$ are
$G$-conjugate and there is an $F$-stable $G$-conjugate $B$ of $A$ such that
the restriction $F\colon B\to B$ is induced by some $w\in
N_{G}(B)/C_{G}(B)^{\circ}$.

  * (b)

If $|N_{G}(A)/C_{G}(A)|=|\operatorname{Aut}(A)|$, then there is a
$G$-conjugate of $A$ in $G^{F}$ if and only if $A$ and $F(A)$ are
$G$-conjugate.

###### Proof.

  * (a)

Suppose $B={}^{g}A\leqslant G^{F}$ for some $g\in G$. Then $B$ is $F$-stable
and $F(A)={}^{F(g^{-1})g}A$ is $G$-conjugate to $A$. The proof of Proposition
4.1 shows that $F$ acts on $F$-stable $G$-conjugates of $B$ as elements of
$N_{G}(B)/C_{G}(B)^{\circ}$ act on $B$. Conversely, suppose $A$ and $F(A)$ are
$G$-conjugate (so $F$-stable $G$-conjugates exist by Proposition 4.1(a)) and
that there is such a $G$-conjugate $B={}^{h}A$ such that $F\colon B\to B$ is
induced by $w\in N_{G}(B)/C_{G}(B)^{\circ}$, that is,
$F(b)=\dot{w}b\dot{w}^{-1}$ for some lift $\dot{w}\in N_{G}(B)$ of $w$. If
$g\in G$ satisfies $g^{-1}F(g)C_{G}(B)^{\circ}=w^{-1}$, then
${}^{g}B=B_{w^{-1}}$ is fixed point-wise by $F$: if $gbg^{-1}\in{}^{g}B$ with
$b\in B$, then
$F(gbg^{-1})=F(g)F(b)F(g)^{-1}=g\dot{w}^{-1}F(b)\dot{w}g^{-1}=gbg^{-1}$, so
${}^{gh}A\leqslant G^{F}$, as claimed.

  * (b)

This follows readily from the previous proof: using the notation of (a), the
assumption on $|\operatorname{Aut}(A)|$ implies that
$N_{G}(B)/C_{G}(B)\cong\operatorname{Aut}(B)$ for every $F$-stable conjugate
of $A$. ∎

###  4.2. Local structure

We continue with the previous notation; moreover, we denote by $wF$ the map
given by $x\mapsto wF(x)w^{-1}$. The following propositions allow us to
determine the local structure of the $G$-conjugates of $A\leqslant G^{F}$ in
$G^{F}$ from the structure of $C_{A}(G)$ and $N_{G}(A)$.

######  Proposition 4.3.

Let $A\leqslant G^{F}$ be a finite subgroup. For $w\in
C_{G}(A)/C_{G}(A)^{\circ}$ let $A_{w}\leqslant G^{F}$ be the $G$-conjugate of
$A$ as defined in Proposition 4.1. If $\dot{w}\in C_{G}(A)$ is any lift of
$w$, then

$(C_{G}(A_{w})^{\circ})^{F}\cong(C_{G}(A)^{\circ})^{\dot{w}F};$

where, as usual, $(C_{G}(A)^{\circ})^{\dot{w}F}$ are the fixed points of
$\dot{w}F$ in $C_{G}(A)^{\circ}$. Furthermore, $(C_{G}(A)^{\circ})^{\dot{w}F}$
is independent of the choice of $\dot{w}$. If $w$ acts as an inner
automorphism of $C_{G}(A)^{\circ}$, then
$(C_{G}(A_{w})^{\circ})^{F}\cong(C_{G}(A)^{\circ})^{F}$.

###### Proof.

Write $C=C_{G}(A)$. By construction, $A_{w}={}^{g}A$ for some $g\in G$ with
$g^{-1}F(g)\in C$ and $w=g^{-1}F(g)C^{\circ}$. Clearly,
$C_{G}(A_{w})^{\circ}={}^{g}(C^{\circ})$. If $c\in C^{\circ}$, then
$F(gcg^{-1})=F(g)F(c)F(g^{-1})=gcg^{-1}$ if and only if
$\dot{w}F(c)\dot{w}^{-1}=c$ for $\dot{w}=g^{-1}F(g)\in C$, hence

$(C_{G}(A_{w})^{\circ})^{F}\cong\\{c\in
C^{\circ}:c=\dot{w}F(c)\dot{w}^{-1}\\}=(C^{\circ})^{\dot{w}F}.$

Since $A$ is closed and $F$-stable, so is $C$. Now [21, Theorem 2.1.2(d)]
proves that the restriction of $F$ to $C$ is a Steinberg endomorphism (in the
sense of [21]), that is, $F|_{C}$ is surjective (hence a bijection since $F$
is an abstract group automorphism, hence injective) and $C^{F}$ is finite. In
particular, $F|_{C}$ is a Steinberg endomorphism in the sense of [32] and as
considered here. It follows from [21, Proposition 1.1.4(c)] that
$F(C^{\circ})\leqslant C^{\circ}$, so $C^{\circ}$ is $F$-stable, and now [32,
Exercise 30.1] shows that $F|_{C^{\circ}}$ is a bijection. In conclusion, $F$
induces a Steinberg automorphism of $C^{\circ}$. If $c\dot{w}$ with $c\in
C^{\circ}$ is another lift of $w$, then the restrictions of $\dot{w}F$ and
$c\dot{w}F$ to $C^{\circ}$ differ by an inner automorphism of $C^{\circ}$, and
[32, Corollary 21.8] shows that $(C^{\circ})^{\dot{w}F}$ and
$(C^{\circ})^{c\dot{w}F}$ are isomorphic, as claimed. In particular, if $w$
induces an inner automorphism on $C^{\circ}$, then the map $wF$ is equal to
$cF$ for some $c\in C^{\circ}$, and
$(C^{\circ})^{wF}=(C^{\circ})^{cF}\cong(C^{\circ})^{F}$. ∎

######  Proposition 4.4.

Let $A\leqslant G^{F}$ be a finite subgroup, let $w\in
C_{G}(A)/C_{G}(A)^{\circ}$, and let $A_{w}\leqslant G^{F}$ be the
$G$-conjugate of $A$ under the correspondence in Proposition 4.1. Then

$\displaystyle N_{G}(A_{w})^{F}/(C_{G}(A_{w})^{\circ})^{F}$
$\displaystyle\cong(N_{G}(A)/C_{G}(A)^{\circ})^{wF},$ $\displaystyle
C_{G}(A_{w})^{F}/(C_{G}(A_{w})^{\circ})^{F}$
$\displaystyle\cong(C_{G}(A)/C_{G}(A)^{\circ})^{wF}.$

###### Proof.

Write $C=C_{G}(A)$ and $N=N_{G}(A)$. By construction, we have $A_{w}=gAg^{-1}$
for some $g\in G$ such that $g^{-1}F(g)C^{\circ}=w$. Let
$\dot{w}=g^{-1}F(g)\in C$. If $n\in N$, then

$\displaystyle\begin{array}[]{lrl}&gng^{-1}&\in
N_{G}(A_{w})^{F}\\\\[2.15277pt]
\iff&gng^{-1}&=F(gng^{-1})=F(g)F(n)F(g)^{-1}=g\dot{w}F(n)\dot{w}^{-1}g^{-1}\\\\[2.15277pt]
\iff&\dot{w}F(n)\dot{w}^{-1}&=n\in N^{\dot{w}F}.\end{array}$

Thus, we have an isomorphism $N_{G}(A_{w})^{F}\cong N^{\dot{w}F}$, which also
induces isomorphisms $C_{G}(A_{w})^{F}\cong C^{\dot{w}F}$,
$(C_{G}(A_{w})^{\circ})^{F}\cong(C^{\circ})^{\dot{w}F}$, and
$(N_{G}(A_{w})^{\circ})^{F}\cong(N^{\circ})^{\dot{w}F}$. Since $G$ is
connected, [21, Theorem 2.1.2(f)] shows that $G^{F}$ and $G^{\dot{w}F}$ are
$G$-conjugate, hence both finite. Since $\dot{w}F$ is also bijective on $G$,
it is a Steinberg morphism (in the sense of [21]). Recall that $N$ and $C$ are
closed and mapped under $\dot{w}F$ into themselves. Now [21, Theorem 2.1.2(d)]
shows that $wF$ induces Steinberg morphisms on $N$ and on $C$, respectively.
Applying [21, Theorem 1.2.1(d)] to $N$ and to $C$, respectively, we see that
$\dot{w}F$ maps $N^{\circ}$ and $C^{\circ}$ onto themselves and that the
natural maps $N^{\dot{w}F}\to(N/N^{\circ})^{\dot{w}F}$ and
$C^{\dot{w}F}\to(C/C^{\circ})^{\dot{w}F}$ are surjective. The claim follows. ∎

###  4.3. The $F$-action on $N_{G}(A)/C_{G}(A)^{\circ}$

The previous results require us to study the action of $F$ on the normaliser
quotient.

######  Proposition 4.5.

Let $A\leqslant G^{F}$ be a finite subgroup. The map $N_{G}(A)\to N_{G}(A)$
induced by $F$ is given by $n\mapsto n\phi(n)$, where $\phi$ is a $1$-cocycle
$N_{G}(A)\to C_{G}(A)$, that is, $\phi$ satisfies

$\phi(n_{1}n_{2})=(\phi(n_{1})^{n_{2}})\phi(n_{2})$

for all $n_{1},n_{2}\in N_{G}(A)$. In particular, $F$ induces the identity map
on $N_{G}(A)/C_{G}(A)$.

###### Proof.

By hypothesis, if $n\in N_{G}(A)$ and $a\in A$ then
$n^{-1}an=F(n^{-1}an)=F(n)^{-1}aF(n)$, thus $F(n)n^{-1}\in C_{G}(A)$ and the
first statement holds. Writing $F(n)=n\phi(n)$, we see that
$n_{1}n_{2}\phi(n_{1}n_{2})=F(n_{1}n_{2})=F(n_{1})F(n_{2})=n_{1}\phi(n_{1})n_{2}\phi(n_{2})$,
from which the cocycle condition on $\phi$ follows. ∎

While $F$ acts trivially on $N_{G}(A)/C_{G}(A)$, it does not necessarily act
trivially on $N_{G}(A)/C_{G}(A)^{\circ}$, even if the action on
$C_{G}(A)/C_{G}(A)^{\circ}$ is trivial:

######  Example 4.6.

Let $G=\operatorname{SL}_{2}(K)$ and $G^{F}=\operatorname{SL}_{2}(3)$, where
$F$ is induced by $x\mapsto x^{3}$. Let $A\leqslant G^{F}$ be a Sylow
2-subgroup and note that $A\unlhd G^{F}$ and $C_{G}(A)=\\{\pm I_{2}\\}$. If
$\alpha$ is a primitive element of $\mathbb{F}_{9}$, then
$m=\operatorname{diag}(\alpha^{2},\alpha^{6})\in N_{G}(A)$ satisfies
$F(m)\cdot I_{2}\cdot m^{-1}=-I_{2}$. This shows that there is only one
$F$-class of $N_{G}(A)$-orbits in $C_{G}(A)/C_{G}(A)^{\circ}$. In particular,
$F$ acts as identity on $N_{G}(A)/C_{G}(A)$, see Proposition 4.5, and on
$C_{G}(A)/C_{G}(A)^{\circ}=C_{G}(A)=\\{\pm I_{2}\\}$, but $F$ acts non-
trivially on $N_{G}(A)/C_{G}(A)^{\circ}$.

If $F$ acts trivially on $N_{G}(A)/C_{G}(A)^{\circ}$, then Propositions 4.3
and 4.4 have the following corollary:

######  Corollary 4.7.

Let $A\leqslant G^{F}$ be a finite subgroup. If $F$ acts trivially on
$N_{G}(A)/C_{G}(A)^{\circ}$, then there is a bijection between the
$N_{G}(A)/C_{G}(A)^{\circ}$-classes in $C_{G}(A)/C_{G}(A)^{\circ}$ and the
$G^{F}$-classes of subgroups of $G^{F}$ that are $G$-conjugate to $A$; this
bijection maps $w\in C_{G}(A)/C_{G}(A)^{\circ}$ to $A_{w}\leqslant G^{F}$, as
defined in the correspondence in Proposition 4.1; moreover

$C_{G^{F}}(A_{w})=(C_{G}(A_{w}))^{F}=(C_{G}(A)^{\circ})^{wF}.C_{C_{G}(A)/C_{G}(A)^{\circ}}(w),$

and

$N_{G^{F}}(A_{w})=(N_{G}(A_{w}))^{F}=(C_{G}(A)^{\circ})^{wF}.C_{N_{G}(A)/C_{G}(A)^{\circ}}(w).$

###  4.4. Cohomology

Recall from [36, I.§5] that if $N$ is a group which acts on a group $C$ by
automorphisms, then the _cohomology set_ $H^{1}(N,C)$ is the set of all
$1$-cocycles, modulo the equivalence relation $\sim$ where $\phi\sim\psi$ if
there exists $c\in C$ such that $\phi(n)=(c^{n})\psi(n)(c^{-1})$ for all $n\in
N$. When $C$ is abelian, this coincides with the usual definition of $H^{1}$
via right-derived functors of Hom. In general, $H^{1}(N,C)$ is not a group,
but a pointed set, the distinguished point $0$ given by the class of the map
$n\mapsto 1$ for all $n\in N$.

The next result shows that for certain $A\leqslant G^{F}$ and given
$N_{G}(A)/C_{G}(A)^{\circ}$ in the algebraic group, we find a $G$-conjugate
$B\leqslant G^{F}$ with the same normaliser quotient in $G^{F}$.

######  Corollary 4.8.

Let $A\leqslant G^{F}$ be a finite subgroup with
$H^{1}(N_{G}(A),C_{G}(A))=\\{0\\}$. There exists a subgroup $B\leqslant G^{F}$
which is $G$-conjugate to $A$, such that

$\displaystyle N_{G}(B)^{F}/(C_{G}(B)^{\circ})^{F}$ $\displaystyle\cong
N_{G}(A)/C_{G}(A)^{\circ},\ \text{and}$ $\displaystyle
C_{G}(B)^{F}/(C_{G}(B)^{\circ})^{F}$ $\displaystyle\cong
C_{G}(A)/C_{G}(A)^{\circ}.$

Moreover, the $F$-classes of $N_{G}(B)/C_{G}(B)^{\circ}$ are precisely the
conjugacy classes of $N_{G}(B)/C_{G}(B)^{\circ}$, and for the $F$-stable
conjugate $B_{w}$ corresponding to the $F$-class of $w\in
C_{G}(B)/C_{G}(B)^{\circ}$ we have

$\displaystyle N_{G}(B_{w})^{F}/(C_{G}(B_{w})^{\circ})^{F}$
$\displaystyle\cong C_{N_{G}(B)/C_{G}(B)^{\circ}}(w),\ \text{and}$
$\displaystyle C_{G}(B_{w})^{F}/(C_{G}(B_{w})^{\circ})^{F}$
$\displaystyle\cong C_{C_{G}(B)/C_{G}(B)^{\circ}}(w).$

###### Proof.

The hypothesis implies that every $1$-cocycle is of the form
$n\mapsto(c^{n})c^{-1}$ for some $c\in C_{G}(A)$. Thus, by Proposition 4.5,
there is $c\in C_{G}(A)$ such that $F\colon N_{G}(A)\to N_{G}(A)$, $n\mapsto
cnc^{-1}$. Now define $B=A_{u}$ as the $F$-stable $G$-conjugate of $A$
corresponding to the $F$-class of $u=c^{-1}C_{G}(A)^{\circ}$, so that the
first claim follows from Proposition 4.4. Recall that we can define
$B=gAg^{-1}$ for some $g\in G$ with
$F(g)C_{G}(A)^{\circ}=gc^{-1}C_{G}(A)^{\circ}$. Since $F(n)=cnc^{-1}$ for all
$n\in N_{G}(A)$ and $F(g)=gc^{-1}\tilde{c}$ for some $\tilde{c}\in
C_{G}(A)^{\circ}$, it follows that every $mC_{G}(B)^{\circ}\in
N_{G}(B)/C_{G}(B)^{\circ}$ is fixed by $F$. Together with Proposition 4.4,
this implies the second claim. ∎

##  5\. Toral subgroups of simple algebraic groups

Recall the notation we have fixed in Section 2. Let $E$ be an elementary
abelian subgroup of $G$ consisting of semisimple elements. Using [22,
(2.13)(iii)], we can assume that $E$ is contained in the normaliser $N_{G}(T)$
of our fixed maximal torus $T$ of $G$. We study separately the cases that $E$
is toral or non-toral: in the former case we assume that $E\leqslant T$; in
the latter case, $E$ has non-trivial image in the Weyl group $W=N_{G}(T)/T$.

Our main tool for classifying toral elementary abelian subgroups of $G$ is the
following proposition, which in fact holds for arbitrary subgroups of $T$. We
later concentrate on elementary abelian subgroups for two reasons: firstly,
methods from linear algebra can be brought to bear, allowing for a very
efficient classification algorithm (see Section 5.1 and [7]); secondly, as
explained in the introduction, elementary abelian subgroups have many
interesting applications, for example, in the classification of maximal
$p$-local and $p$-radical subgroups.

######  Proposition 5.1.

Let $G$ be a simple algebraic group, with maximal torus $T$ and Weyl group
$W$. If $A,B\leqslant T$ are finite subgroups, then the following hold.

  * (a)

If $A=B^{g}$ with $g\in G$, then $g=vc$ for some $v\in N_{G}(T)$ and $c\in
C_{G}(A)^{\circ}$; in particular, $A$ and $B$ are conjugate in $N_{G}(T)$.

  * (b)

We can decompose $N_{G}(A)\cong
C_{G}(A)^{\circ}.(C_{G}(A)/C_{G}(A)^{\circ}).(N_{G}(A)/C_{G}(A))$, with
isomorphisms

$C_{G}(A)/C_{G}(A)^{\circ}\cong
C_{W}(A)/W(C_{G}(A)^{\circ})\quad\text{and}\quad N_{G}(A)/C_{G}(A)\cong
N_{W}(A)/C_{W}(A),$

where $W(C_{G}(A)^{\circ})$ is the Weyl group of the reductive group
$C_{G}(A)^{\circ}$.

###### Proof.

  * (a)

Note that $A\leqslant T\cap T^{g}$. If $a\in A$, then $a$ is semisimple,
and[32, Theorem 14.2] shows that $C_{{G}}(a)^{\circ}$ is a connected reductive
group containing ${T}$ and $T^{g}$. Since $A$ is finite, induction proves that
$C_{{G}}(A)^{\circ}$ contains $T$ and $T^{g}$; these are therefore
$C_{G}(A)^{\circ}$-conjugate, hence ${T}={T}^{gc}$ for some $c\in
C_{G}(A)^{\circ}$, that is, $gc\in N_{G}({T})$.

  * (b)

In the following we abbreviate $C=C_{G}(A)$; clearly,
$N_{G}(A)=C^{\circ}.(C/C^{\circ}).(N_{G}(A)/C)$. It follows from Part a) that
$N_{G}(A)=N_{N_{G}(T)}(A)C^{\circ}=N_{N_{G}(T)}(A)C$, and so

$\displaystyle N_{G}(A)/C$ $\displaystyle\cong$ $\displaystyle
N_{N_{G}(T)}(A)/(N_{N_{G}(T)}(A)\cap C)$ $\displaystyle\cong$
$\displaystyle(N_{N_{G}(T)}(A)/T)/(C_{N_{G}(T)}(A)/T)$ $\displaystyle=$
$\displaystyle N_{W}(A)/C_{W}(A).$

Note that

$\displaystyle N_{N_{G}(T)}(A)C^{\circ}/C^{\circ}$ $\displaystyle\cong$
$\displaystyle N_{N_{G}(T)}(A)/(N_{N_{G}(T)}(A)\cap C^{\circ})$
$\displaystyle\cong$ $\displaystyle N_{W}(A)/(N_{C^{\circ}}(T)/T)$
$\displaystyle=$ $\displaystyle N_{W}(A)/W(C^{\circ}).$

These isomorphisms allow us to identify $C/C^{\circ}\leqslant
N_{N_{G}(T)}(A)C^{\circ}/C^{\circ}$ as a subgroup of $N_{W}(A)/W(C^{\circ})$,
and we deduce $C/C^{\circ}\cong
C_{N_{W}(A)/W(C^{\circ})}(A)=C_{W}(A)/W(C^{\circ})$. ∎

###  5.1. Computation of toral elementary abelian subgroups

Continuing with the previous notation, we now explain how to classify, up to
conjugacy, the toral elementary abelian $p$-subgroups of $G$ algorithmically,
by working in a suitably-chosen finite group of Lie type. For technical
reasons, we first assume that $p$ is odd, and treat the case $p=2$ in a
moment.

Recall that $T$ is a fixed maximal torus of $G$ with corresponding set of
roots $\Phi$. First, by Dirichlet’s Theorem we can choose a prime-power $q$
such that $p$ divides $q-1$; by Proposition 3.1, we may assume that the
characteristic $\ell$ divides $q$. Second, we choose a Steinberg morphism $F$
of $G$ which induces the $q$-power map on $T$. This gives rise to the finite
group $G^{F}$ of Lie type, containing the split torus $T^{F}$, which is a
product of cyclic groups of order $q-1$. This latter group contains the
subgroup $T_{(p)}$ defined as $T_{(p)}=\Omega_{1}(O_{p}(T))$, and thus
contains a representative of each toral elementary abelian $p$-subgroup of
$G$.

Recall from Proposition 5.1 that two toral subgroups of $T$ are conjugate in
$G$ if and only if they are conjugate under the action of the Weyl group
$W=N_{G}(T)/T$. In $G^{F}$ we therefore construct the _extended Weyl group_
$V$ of $G^{F}$, see [38, Notation 2.1]. Originally defined by Tits for the
simply connected group, $V\leqslant N_{G}(T)$ is an extension of $W$ by an
elementary abelian subgroup of order $\gcd(2,\ell-1)^{r}$, where $r$ is the
rank of $G$, generated by a certain finite set of elements
$\\{n_{\alpha}(1):\alpha\in\Phi\\}$, such that $V$ maps onto $W$ under the
natural projection map $N_{G}(T)\to W$. In particular, $V$ normalises $T^{F}$,
and $VT^{F}/T^{F}$ is isomorphic to $W$. Our choice of $F$ and [14, Section
12.1 and p. 61] imply that $V\leqslant G^{F}$, and therefore the action of $W$
on $T_{(p)}$ can be viewed inside the finite group $VT^{F}\leqslant G^{F}$. In
our case, $G$ is not necessarily simply connected, but if $G=G_{{\text{\rm
sc}}}/Z$ then we construct $VZ/Z$ as its extended Weyl group. We compute
$\tilde{W}\leqslant\operatorname{GL}_{r}(p)$ as the image of $V$ defined by
its action on $T_{(p)}$, and we determine all toral elementary abelian
$p$-subgroups of $G$, up to conjugacy, by computing the $\tilde{W}$-conjugacy
classes of elementary abelian subgroups contained in $T_{(p)}$. In the
following, let $E$ be such an elementary abelian subgroup; we now describe how
to determine the structure of $N_{G}(E)$ within $G^{F}$.

By Proposition 5.1(b), to describe the structure of $N_{G}(E)$ we need to
determine $C_{G}(E)^{\circ}$ and the finite groups
$C_{W}(E)/W(C_{G}(E)^{\circ})$ and $N_{W}(E)/C_{W}(E)$. As in the proof of
Proposition 5.1(a), it follows from [32, Theorem 14.2] and a finite inductive
argument that $C_{G}(E)^{\circ}$ is generated by $T$ and those root subgroups
$U_{\alpha}=\\{x_{\alpha}(c):c\in K\\}$ which commute with $E$. To find which
root subgroups we need, it suffices to check which of the finitely many
elements $x_{\alpha}(1)$ commute with $E$; note that each $x_{\alpha}(1)$ lies
in $G^{F}$ by our choice of $F$. This allows us to compute the root datum of
$C_{G}(E)^{\circ}$. Since $T\leqslant C_{G}(E)^{\circ}$, adding a suitable
toral subgroup determined most of the structure of $C_{G}(E)^{\circ}$; the
remaining work is to determine the isogeny types of the occurring simple
factors in $C_{G}(E)^{\circ}$. It now remains to understand
$C_{W}(E)/W(C_{G}(E)^{\circ})$ and $N_{W}(E)/C_{W}(E)$. The action of
$W=N_{G}(T)/T$ on $T$ restricts to an action on the characteristic subgroup
$T_{(p)}$, giving a surjective homomorphism $W\to\tilde{W}$. The only
algebraic endomorphisms of the multiplicative group $K^{\times}$ are power
maps, and it follows that the full group of algebraic automorphisms of $T$ is
isomorphic to $\operatorname{GL}_{r}(\mathbb{Z})$. Moreover, the restriction
to $T_{(p)}$ is precisely the homomorphism
$\operatorname{GL}_{r}(\mathbb{Z})\to\operatorname{GL}_{r}(p)$ given by
reducing matrix entries modulo $p$. Restricting this map to the finite
subgroup $W\leqslant\operatorname{Aut}(T)$ gives us the above map
$W\to\tilde{W}$. A result of Minkowski [33], see also [37, Lemma 1], tells us
that the kernel of
$\operatorname{GL}_{r}(\mathbb{Z})\to\operatorname{GL}_{r}(\mathbb{Z}/m\mathbb{Z})$
is torsion-free for all integers $m>2$, thus the induced map $W\to\tilde{W}$
is injective. This proves that we can compute the entire structure of
$N_{W}(E)$ inside $\tilde{W}$; in conclusion, the structure of $N_{G}(E)$ is
determined.

When $p=2$, the above argument goes through similarly, with the exception that
Minkowski’s lemma fails to apply to the map
$\operatorname{GL}_{r}(\mathbb{Z})\to\operatorname{GL}_{r}(\mathbb{Z}/2\mathbb{Z})$;
indeed, the kernel can contain elements of order $2$ in this case. For this
reason, instead of working with $T_{(2)}$, we pick $q$ so that $4\mid q-1$,
and let $T_{(4)}=\Omega_{2}(O_{2}(T))$ be the characteristic subgroup of $T$
generated by elements of order dividing $4$. Then the $W$-orbits on elementary
abelian $2$-subgroups of $T$ coincide with the $\tilde{W}$-orbits on
elementary abelian $2$-subgroups of $T_{(4)}$, where $\tilde{W}$ is the image
of $W$ in
$\operatorname{Aut}(T_{(4)})=\operatorname{GL}_{r}(\mathbb{Z}/4\mathbb{Z})$.
We can now apply Minkowski’s lemma as above to deduce that the map
$W\to\tilde{W}$ is an isomorphism. This proves that for each elementary
abelian $2$-subgroup $E\leqslant T$, the structure of $N_{G}(E)$ is again
determined by working in the finite group $N_{\tilde{W}}(E)$.

A Magma implementation of this algorithm is available at [7].

###  5.2. Translation to finite groups

We use the notation of Section 4, that is, $G$ is a connected reductive
algebraic group over the algebraically closed field $K$ of characteristic
$\ell$, with maximal torus $T$ and Weyl group $W$. The prime $p$ is different
from $\ell$, and $F$ is a (possibly twisted) Steinberg endomorphism $G$ such
that $T$ is $F$-stable. The aim of this section is to describe how to
classify, up to $G^{F}$-conjugacy, the toral elementary abelian subgroups of
$G^{F}$, together with their local structure in $G^{F}$, assuming that this
information is known for $G$. The approach is based on the following lemma
(and results of Section 4), and summarised in the subsequent remark.

######  Lemma 5.2.

If $A\leqslant T\cap G^{F}$ is toral, then

$N_{G}(A)/C_{G}(A)^{\circ}=N_{G^{F}}(A)/(C_{G}(A)^{\circ})^{F}\quad\text{and}\quad
C_{G}(A)/C_{G}(A)^{\circ}=C_{G^{F}}(A)/(C_{G}(A)^{\circ})^{F},$

and $F$ acts trivially on $N_{G}(A)/C_{G}(A)^{\circ}$.

###### Proof.

The kernel of the restriction to $N_{G^{F}}(A)$ of the natural projection
$N_{G}(A)\to N_{G}(A)/C_{G}(A)^{\circ}$ is $C_{G}(A)^{\circ}\cap
G^{F}=(C_{G}(A)^{\circ})^{F}$; thus, the first claim follows if we show that
this restriction is surjective. For this recall that the extended Weyl group
$V\leqslant G^{F}$ satisfies $N_{G}(T)=\langle T,V\rangle$, see Section 5.1.
Moreover, Proposition 5.1 shows that if $g\in N_{G}(A)$, then $g=vc$ for some
$v\in N_{G}(T)$ and $c\in C_{G}(A)^{\circ}$. Since $N_{G}(T)=\langle
V,T\rangle$ and $T\leqslant C_{G}(A)^{\circ}$, we can assume that $v\in
N_{G^{F}}(A)$ and $c\in C_{G}(A)^{\circ}$; this shows that $N_{G^{F}}(A)\to
N_{G}(A)/C_{G}(A)^{\circ}$ is surjective. The claim about the centralisers
follows analogously. ∎

######  Remark 5.3.

For a simple algebraic group $G$, to classify all toral elementary abelian
$p$-subgroups of $G^{F}$ up to conjugacy, we now have the following recipe.

  * (a)

Determine the toral elementary abelian $p$-subgroups $A\leqslant G$ and their
local structure, up to $G$-conjugacy. This can be done with the method
described in Section 5.1. For each $G$-class, determine a $G$-class
representative $A$ that lies in $G^{F}$ (if such a representative exists);
Corollary 4.2 is useful here.

  * (b)

For each such representative $A\leqslant G^{F}$, determine the
$N_{G}(A)/C_{G}(A)^{\circ}$-classes in $C_{G}(A)/C_{G}(A)^{\circ}$.

  * (c)

For each $N_{G}(A)/C_{G}(A)^{\circ}$-class representative $w\in
C_{G}(A)/C_{G}(A)^{\circ}$, determine the local structure of $A_{w}\leqslant
G^{F}$ using Corollary 4.7; note that this corollary is applicable due to
Lemma 5.2.

In Step (b), we note that if $C_{G}(A)=C_{G}(A)^{\circ}$, then the $G$-class
of $A$ contains only one $G^{F}$-class, and $C_{G^{F}}(A)=C_{G}(A)^{F}$ and
$N_{G^{F}}(A)=C_{G}(A)^{F}.(N_{G}(A)/C_{G}(A))$. If $C_{G}(A)\neq
C_{G}(A)^{\circ}$, then $p$ is a torsion prime by Theorem 3.4, and some work
is required in Step (c).

##  6\. Non-toral subgroups of exceptional algebraic groups

We now turn our attention to non-toral elementary abelian subgroups. In this
case, the analogue of Proposition 5.1 fails, that is, the normaliser structure
of such a subgroup is not controlled by the normaliser of a maximal torus, and
thus more ad-hoc calculations are required. Nevertheless the results of
Section 3.1 still apply, allowing us to transfer many known results from the
characteristic $0$ setting.

Let $G$ be an exceptional simple algebraic group, still over an algebraically
closed field $K$ of characteristic $\ell$, different from a fixed prime $p$.
Each non-toral elementary abelian $p$-subgroup of $G$ is contained in a
maximal such subgroup, which have been classified up to conjugacy by Griess
[22] for groups over $\mathbb{C}$. Also for complex groups, when $p$ is odd, a
complete description of non-toral elementary abelian $p$-subgroups and their
normaliser structure is given by [11, Section 8]. When $p=2$ and $G$ is
adjoint, much information regarding the collection of _all_ elementary abelian
$2$-subgroups is given in [40]. Comparing this with the information on toral
subgroups provided by our algorithm, this allows us to derive a complete list
of non-toral subgroups, and it then remains to consider the normaliser
structure of these subgroups.

###  6.1. Statement of results for odd $p$

In view of Proposition 3.1 and Lemma 3.3, the next proposition follows from
[22, §1] and [11, §8]. The class distribution of the non-toral subgroup in
$F_{4}$ is derived in the proof of [22, Theorem 7.4].

######  Proposition 6.1 (Griess; Andersen et al.).

Let $p$ be an odd prime and let $G$ be a simple algebraic group of exceptional
type over an algebraically closed field $K$ of characteristic $\ell\neq p$. If
$E\leqslant G$ is a non-toral elementary abelian $p$-subgroup, then
$p\in\\{3,5\\}$ and $(G,E)$ appears in Table 3. Each line of the table
corresponds to a unique $G$-conjugacy class; supplementing comments are given
in Remark 6.2.

Table 3. Non-toral elementary abelian $p$-subgroups of exceptional algebraic groups, $p$ odd $\boldsymbol{G}$ | $\boldsymbol{E}$ | $\boldsymbol{\operatorname{Dist}(E)}$ | $\boldsymbol{C_{G}(E)}$ | $\boldsymbol{N_{G}(E)}$
---|---|---|---|---
$E_{8}$ | $5^{3}$ | $5{\rm C}_{124}$ | $5^{3}$ | $5^{3}\mathpunct{:}\operatorname{SL}_{3}(5)$
$F_{4}$ | $3^{3}$ | $3{\rm C}_{26}$ | $3^{3}$ | $3^{3}\mathpunct{:}\operatorname{SL}_{3}(3)$
$E_{6,{\text{\rm sc}}}$ | $3^{4}$ | $3{\rm C}_{78}^{\phantom{1}}{\rm E}_{1}^{\phantom{1}}{\rm E}^{\prime}_{1}$ | $3^{4}$ | $3^{1+3+3}\mathpunct{:}\operatorname{SL}_{3}(3)$
2-5  | $3^{3}$ | $3{\rm C}_{26}$ | $3^{4}$ | $3\times(3^{3}\mathpunct{:}\operatorname{SL}_{3}(3))$
$E_{6,{\text{\rm ad}}}$ | $(3^{4})_{a}$ | $3{\rm C}_{62}{\rm D}_{18}$ | $(3^{4})_{a}$ | $(3^{4}).((3^{2}\times 3^{2})\mathpunct{:}(3\times\operatorname{GL}_{2}(3)))$
| $(3^{4})_{b}$ | $3{\rm AB}_{6}{\rm C}_{50}{\rm D}_{24}$ | $3^{2}\times T_{2}$ | $(3^{2}\times T_{2}).(3^{2}\mathpunct{:}(\operatorname{Dih}_{12}\times\operatorname{SL}_{2}(3)))$
2-5  | $(3^{3})_{a}$ | $3{\rm C}_{20}{\rm D}_{6}$ | $3^{2}\times(T_{2}:3)$ | $(3^{2}\times(T_{2}:3)).(3^{1+2}_{+}:2^{2})$
| $(3^{3})_{b}$ | $3{\rm C}_{26}$ | $(3^{3})_{b}.(3^{3})$ | $(3^{3})_{b}.(3^{3}\mathpunct{:}\operatorname{SL}_{3}(3))$
| $(3^{3})_{c}$ | $3{\rm C}_{2}{\rm D}_{24}$ | $3^{2}\times\operatorname{SL}_{3}(K)$ | $(3^{2}\times\operatorname{SL}_{3}(K)).(3^{2}\mathpunct{:}(2\times\operatorname{SL}_{2}(3)))$
| $(3^{3})_{d}$ | $3{\rm AB}_{2}{\rm C}_{16}{\rm D}_{8}$ | $3^{2}\times\operatorname{GL}_{2}(K)$ | $(3^{2}\times\operatorname{GL}_{2}(K)).(2\times\operatorname{SL}_{2}(3))$
2-5  | $(3^{2})_{a}$ | $3{\rm C}_{6}{\rm D}_{2}$ | $3^{2}\times\operatorname{PSL}_{3}(K)$ | $(3^{2}\times\operatorname{PSL}_{3}(K)).6$
| $(3^{2})_{b}$ | $3{\rm D}_{8}$ | $3^{2}\times G_{2}(K)$ | $(3^{2}\times G_{2}(K)).\operatorname{SL}_{2}(3)$
$E_{7}$ | $3^{4}$ | $3{\rm B}_{2}{\rm C}_{78}$ | $(3^{3}\times T_{1})$ | $(3^{3}\times T_{1}).((3^{3}.\operatorname{SL}_{3}(3))\mathpunct{:}2)$
2-5  | $3^{3}$ | $3{\rm C}_{26}$ | $(3^{3}\times\operatorname{SL}_{2}(K))/Z$ | $(3^{3}\mathpunct{:}\operatorname{SL}_{3}(3))\times\operatorname{SL}_{2}(K)/Z$
$E_{8}$ | $(3^{5})_{a}$ | $3{\rm A}_{156}{\rm B}_{80}{\rm D}_{6}$ | $3^{3}\times T_{2}$ | $(3^{3}\times T_{2}).(3^{4}\mathpunct{:}(2^{2}\times\operatorname{SL}_{3}(3)))$
| $(3^{5})_{b}$ | $3{\rm A}_{162}{\rm B}_{80}$ | $3^{5}$ | $3^{5}.(3^{4}\mathpunct{:}{\rm Sp}_{4}(3).2)$
2-5  | $(3^{4})_{a}$ | $3{\rm A}_{52}{\rm B}_{26}{\rm D}_{2}$ | $3^{3}\times\operatorname{GL}_{2}(K)$ | $(3^{3}\times\operatorname{GL}_{2}(K)).(\operatorname{SL}_{3}(3)\times 2)$
| $(3^{4})_{b}$ | $3{\rm A}_{54}{\rm B}_{26}$ | $3^{3}\times(T_{2}:3)$ | $(3^{3}\times(T_{2}:3)).(3^{1+4}\mathpunct{:}(2\times\operatorname{GL}_{2}(3)))$
| $(3^{4})_{c}$ | $3{\rm B}_{80}$ | $3^{3}\times\operatorname{SL}_{3}(K)$ | $(3^{3}\times\operatorname{SL}_{3}(K)).(3^{3}:(2\times\operatorname{SL}_{3}(3)))$
2-5  | $(3^{3})_{a}$ | $3{\rm A}_{18}{\rm B}_{8}$ | $3^{3}\times\operatorname{PSL}_{3}(K)$ | $(3^{3}\times\operatorname{PSL}_{3}(K)).(3^{2}\mathpunct{:}\operatorname{GL}_{2}(3))$
| $(3^{3})_{b}$ | $3{\rm B}_{26}$ | $3^{3}\times G_{2}(K)$ | $(3^{3}\times G_{2}(K)).\operatorname{SL}_{3}(3)$

######  Remark 6.2.

These remarks supplement Proposition 6.1.

  * (a)

The normaliser structure is taken from [11, §8], which correct errors in [22]
for the centralisers and normalisers of the non-toral subgroups $3^{3}$ in
$E_{7}$, and $(3^{5})_{a}$ and $(3^{3})_{b}$ in $E_{8}$. Although we only give
normaliser structure using short notation, in [11] the precise action of
$N_{G}(E)/C_{G}(E)$ is given on each subgroup $E$.

  * (b)

For adjoint $G$ of type $E_{6}$, in [11, Theorem 8.10] the distribution is
given of the pre-image $\widetilde{E}$ of $E$ in the simply connected cover of
$G$. Since multiplication by an element of the centre permutes $3{\rm A}$,
$3{\rm B}$, and elements whose square is in $3{\rm B}$ (while preserving the
class of elements in $3{\rm C}$ and $3{\rm D}$), it is straightforward to
derive the above distribution of $E$ from that of $\widetilde{E}$, and vice-
versa. The notation $3{\rm AB}$ denotes the unique conjugacy class which is
the image of the classes $3{\rm A}$ and $3{\rm B}$ (and also the inverse of
elements in $3{\rm B}$). The inverse of the class $3{\rm E}$ is denoted $3{\rm
E}^{\prime}$.

  * (c)

For $G$ of type $E_{7}$, the classification is independent of the isogeny
type, because the simply connected group has centre of order $2$, and thus by
the Schur-Zassenhaus theorem, for $p\neq 2$ each elementary abelian
$p$-subgroup of the adjoint group lifts uniquely to an isomorphic copy in the
simply connected group. In the table, the group $Z$ is the kernel of the
isogeny $G_{{\text{\rm sc}}}\to G$ from the simply connected cover
$G_{{\text{\rm sc}}}$ of $G$; thus $Z=1$ if $G=G_{{\text{\rm sc}}}$ and $Z=2$
if $G=G_{{\text{\rm ad}}}$.

###  6.2. Statement of results for $p=2$

We now consider elementary abelian $2$-subgroups. Again, by Proposition 3.1
and Lemma 3.3 we are able to use results concerning complex Lie groups and
compact real Lie groups, in particular [22] and [40]. However, in this case
complete information is not given, and we supplement existing results with
arguments for $E_{7,{\text{\rm sc}}}$, as well as our own normaliser structure
calculations. Since the lists of subgroups for $E_{8}$ and $E_{7,{\text{\rm
ad}}}$ are much larger than the other cases, we separate these out for
readability. For the same reason, we have outsourced some more technical
considerations to the appendix.

######  Proposition 6.3.

Let $G$ be a simple algebraic group of exceptional type over an algebraically
closed field $K$ of characteristic $\ell\neq 2$. If $E\leqslant G$ is a non-
toral elementary abelian $2$-subgroup, then $(G,E)$ appears in Table 5, Table
6, or Table 7. Each line of a table corresponds to a unique $G$-conjugacy
class of subgroups; supplementing comments are given in Remark 6.4.

######  Remark 6.4.

These remarks supplement Proposition 6.3.

  * (a)

For $G$ of type $E_{6}$, the classification is independent of the isogeny
type, because the simply connected group has centre of order $3$, and thus by
the Schur-Zassenhaus theorem, for $p\neq 3$ each elementary abelian
$p$-subgroup of the adjoint group lifts uniquely to an isomorphic copy in the
simply connected group. In the table, the group $Z$ is the kernel of the
isogeny $G_{{\text{\rm sc}}}\to G$ from the simply connected cover
$G_{{\text{\rm sc}}}$ of $G$; thus $Z=1$ if $G=G_{{\text{\rm sc}}}$ and $Z=3$
if $G=G_{{\text{\rm ad}}}$.

  * (b)

For $G$ adjoint of type $E_{7}$, the notation $2{\rm BC}$ denotes the unique
conjugacy class which is the image of the classes $2{\rm B}$ and $2{\rm C}$,
and we abuse notation and write $4{\rm A}$ and $4{\rm H}$ for the involutions
coming from the corresponding classes in the simply connected cover of $G$.

  * (c)

Since much of the information in Tables 6 and 7 is derived from [40, Sections
2–8], in the third column we give the name used in the classification derived
there; we refer to Appendices A.1 and B.1 for details. The notation of the
centralisers in Tables 6 and 7 is as in [21].

The remainder of this section is devoted to a proof of Proposition 6.3. Using
the algorithm of Section 5.1, we find that the number of classes of toral
elementary abelian $2$-subgroups is as given in Table 4.

Table 4. Number of classes of toral elementary abelian $2$-subgroups, where the bracketed numbers count subgroups of order $2^{0}$, $2^{1}$, etc. $G$ | $G_{2}$ | $F_{4}$ | $E_{6}$ | $E_{7}$ | $E_{8}$
---|---|---|---|---|---
Number | $(1,1,1)$ | $(1,2,3,2,1)$ | $(1,2,4,4,4,2,1)$ | $(1,3,5,7,7,5,3,1)$ | $(1,2,4,5,7,5,4,2,1)$
Total | $3$ | $9$ | $18$ | $32$ | $31$
Table 5. Non-toral elementary abelian $2$-subgroups of algebraic groups $G_{2}$, $F_{4}$, $E_{6}$ and $E_{7,{\text{\rm sc}}}$. $\boldsymbol{G}$ | $\boldsymbol{E}$ | $\boldsymbol{\operatorname{Dist}(E)}$ | $\boldsymbol{C_{G}(E)}$ | $\boldsymbol{N_{G}(E)}$
---|---|---|---|---
$G_{2}$ | $2^{3}$ | $2{\rm A}_{7}$ | $2^{3}$ | $2^{3}.\operatorname{SL}_{3}(2)$
$F_{4}$ | $2^{5}$ | $2{\rm A}_{28}{\rm B}_{3}$ | $2^{5}$ | $2^{5}.(2^{2\cdot 3}).[\operatorname{SL}_{3}(2)\times\operatorname{Sym}_{3}]$
2-5  | $2^{4}$ | $2{\rm A}_{14}{\rm B}_{1}$ | $2^{3}\times T_{1}.2$ | $(2^{3}\times T_{1}.2).(2^{3}\mathpunct{:}\operatorname{SL}_{3}(2))$
2-5  | $2^{3}$ | $2{\rm A}_{7}$ | $2^{3}\times\operatorname{PGL}_{2}(K)$ | $(2^{3}\mathpunct{:}\operatorname{SL}_{3}(2))\times\operatorname{PGL}_{2}(K)$
$E_{6}$ | $2^{5}$ | $2{\rm A}_{28}{\rm B}_{3}$ | $2^{3}\times T_{2}$ | $(2^{3}\times T_{2}).(2^{2\cdot 3}).(\operatorname{Sym}_{3}\times\operatorname{SL}_{3}(2))$
2-5  | $2^{4}$ | $2{\rm A}_{14}{\rm B}_{1}$ | $2^{3}\times\operatorname{GL}_{2}(K)$ | $(2^{3}\times\operatorname{GL}_{2}(K)).(2^{3}\mathpunct{:}\operatorname{SL}_{3}(2))$
2-5  | $2^{3}$ | $2{\rm A}_{7}$ | $2^{3}\times\operatorname{SL}_{3}(K)/Z$ | $(2^{3}\mathpunct{:}\operatorname{SL}_{3}(2))\times\operatorname{SL}_{3}(K)/Z$
$E_{7,{\text{\rm sc}}}$ | $2^{6}$ | $2{\rm A}_{1}{\rm B}_{31}{\rm C}_{31}$ | $2^{3}\times\operatorname{SL}_{2}(K)^{3}$ | $C_{G}(E).2^{2\cdot 3}.(\operatorname{SL}_{3}(2)\times\operatorname{Sym}_{3})$
2-5  | $(2^{5})_{a}$ | $2{\rm A}_{1}{\rm B}_{15}{\rm C}_{15}$ | $2^{3}\times\operatorname{SL}_{2}(K)\times{\rm Sp}_{4}(K)$ | $C_{G}(E).(2^{3}\mathpunct{:}\operatorname{SL}_{3}(2))$
| $(2^{5})_{b}$ | $2{\rm B}_{28}{\rm C}_{3}$ | $2^{3}\times\operatorname{SL}_{2}(K)^{3}$ | $C_{G}(E).2^{2\cdot 3}.(\operatorname{SL}_{3}(2)\times\operatorname{Sym}_{3})$
| $(2^{5})_{c}$ | $2{\rm B}_{16}{\rm C}_{15}$ | $2^{3}\times\operatorname{SL}_{2}(K)^{3}$ | $C_{G}(E).(2\times(2^{3}\mathpunct{:}\operatorname{SL}_{3}(2)))$
| $(2^{5})_{d}$ | $2{\rm B}_{12}{\rm C}_{19}$ | $2^{3}\times\operatorname{SL}_{2}(K)^{3}$ | $C_{G}(E).(2^{2+6}\mathpunct{:}(\operatorname{Sym}_{3}\times\operatorname{Sym}_{3}))$
2-5  | $(2^{4})_{a}$ | $2{\rm A}_{1}{\rm B}_{7}{\rm C}_{7}$ | $2^{3}\times{\rm Sp}_{6}(K)$ | $(2^{3}\mathpunct{:}\operatorname{SL}_{3}(2))\times{\rm Sp}_{6}(K)$
| $(2^{4})_{b}$ | $2{\rm B}_{14}{\rm C}_{1}$ | $2^{3}\times\operatorname{SL}_{2}(K)\times{\rm Sp}_{4}(K)$ | $C_{G}(E).(2^{3}\mathpunct{:}\operatorname{SL}_{3}(2))$
| $(2^{4})_{c}$ | $2{\rm B}_{8}{\rm C}_{7}$ | $2^{3}\times\operatorname{SL}_{2}(K)\times{\rm Sp}_{4}(K)$ | $(2^{3}\mathpunct{:}\operatorname{SL}_{3}(2))\times\operatorname{SL}_{2}(K)\times{\rm Sp}_{4}(K)$
| $(2^{4})_{d}$ | $2{\rm B}_{6}{\rm C}_{9}$ | $2^{3}\times\operatorname{SL}_{2}(K)\times{\rm Sp}_{4}(K)$ | $C_{G}(E).(2_{+}^{1+4}\mathpunct{:}\operatorname{Sym}_{3})$
2-5  | $(2^{3})_{a}$ | $2{\rm B}_{7}$ | $2^{3}\times{\rm Sp}_{6}(K)$ | $(2^{3}\mathpunct{:}\operatorname{SL}_{3}(2))\times{\rm Sp}_{6}(K)$
| $(2^{3})_{b}$ | $2{\rm B}_{3}{\rm C}_{4}$ | $2^{3}\times{\rm Sp}_{6}(K)$ | $(2^{3}\mathpunct{:}2^{2}.\operatorname{Sym}_{3})\times{\rm Sp}_{6}(K)$
Table 6. Non-toral elementary abelian $2$-subgroups of $E_{7,{{\text{\rm ad}}}}$, see Table 8 for more details on the normaliser quotient; here $Z$ is the kernel of $E_{7,{{\text{\rm sc}}}}\to E_{7,{{\text{\rm ad}}}}$. The notation $\leftrightarrow$ denotes an involution swapping two isomorphic simple subgroup factors. The subgroups listed in column labelled “$U$” are used in Case (2) of the proof of Proposition 6.6. $\boldsymbol{E}$ | $\boldsymbol{\operatorname{Dist}(E)}$ | Name | $\dim$ | $\boldsymbol{U}$ | $\boldsymbol{C_{G}(E)}$ | $\boldsymbol{N_{G}(E)/C_{G}(E)}$
---|---|---|---|---|---|---
$2^{8}$ | $2{\rm BC}_{63}4{\rm A}_{164}4{\rm H}_{28}$ | $F_{0,1,0,2}$ | $0$ | $F_{0,1,1,1}$ | $2^{8}$ | $2^{7}\mathpunct{:}\operatorname{SO}_{7}(2)$
$(2^{7})_{a}$ | $2{\rm BC}_{31}4{\rm A}_{84}4{\rm H}_{12}$ | $F_{2,3}$ | $0$ | $F_{1,3}$ | $2^{7}$ | $(2^{2\cdot 2}\times 2^{2\cdot 3})\mathpunct{:}(\operatorname{Sym}_{3}\times\operatorname{Sym}_{3}\times\operatorname{SL}_{3}(2))$
$(2^{7})_{b}$ | $2{\rm BC}_{31}4{\rm A}_{84}4{\rm H}_{12}$ | $F_{0,1,1,1}$ | $0$ | $F_{0,1,0,1}$ | $2^{8}$ | $(2^{6}\mathpunct{:}2^{5})\mathpunct{:}\operatorname{SO}_{5}(2)$
$(2^{7})_{c}$ | $2{\rm BC}_{31}4{\rm A}_{80}4{\rm H}_{16}$ | $F_{1,0,0,2}$ | $1$ | $F_{1,2}^{\prime\prime}$ | $2^{6}\times T_{1}.i$ | $(2^{6}\mathpunct{:}2^{4})\mathpunct{:}\operatorname{SO}_{5}(2)$
$(2^{7})_{d}$ | $2{\rm BC}_{63}4{\rm A}_{64}$ | $F_{0,3}^{\prime\prime}$ | $0$ | $F_{1,2}^{\prime\prime}$ | $2^{8}$ | $2^{6}\mathpunct{:}{\rm Sp}_{6}(2)$
$(2^{6})_{a}$ | $2{\rm BC}_{15}4{\rm A}_{42}4{\rm H}_{6}$ | $F_{1,3}$ | $1$ | $F_{0,3}$ | $2^{5}\times T_{1}.i$ | $(2^{2}\times 2^{3})\mathpunct{:}(\operatorname{Sym}_{3}\times\operatorname{SL}_{3}(2))$
$(2^{6})_{b}$ | $2{\rm BC}_{15}4{\rm A}_{36}4{\rm H}_{12}$ | $F_{2,2}$ | $4$ | $F_{2,1}$ | $2^{2}\times T_{4}.2$ | $(2^{2\cdot 2}\times 2^{2\cdot 2})\mathpunct{:}(\operatorname{Sym}_{3}\times\operatorname{Sym}_{3}\times\operatorname{Sym}_{3})$
$(2^{6})_{c}$ | $2{\rm BC}_{31}4{\rm A}_{28}4{\rm H}_{4}$ | $F_{2,3}^{\prime}$ | $3$ | $F_{1,3}^{\prime}$ | $2^{3}\times T_{3}.2$ | $(2^{2\cdot 1}\times 2^{2\cdot 3})\mathpunct{:}(\operatorname{Sym}_{3}\times\operatorname{SL}_{3}(2))$
$(2^{6})_{d}$ | $2{\rm BC}_{15}4{\rm A}_{38}4{\rm H}_{10}$ | $F_{0,0,0,2}$ | $3$ | $F_{0,2}^{\prime\prime}$ | $2^{6}\times\operatorname{PSL}_{2}(K)$ | $2^{5}\mathpunct{:}\operatorname{SO}_{5}(2)$
$(2^{6})_{e}$ | $2{\rm BC}_{15}4{\rm A}_{42}4{\rm H}_{6}$ | $F_{0,1,0,1}$ | $1$ | $F_{0,2}^{\prime\prime}$ | $2^{6}\times(T_{1}.i)$ | $2^{5}\mathpunct{:}\operatorname{SO}_{5}(2)$
$(2^{6})_{f}$ | $2{\rm BC}_{15}4{\rm A}_{44}4{\rm H}_{4}$ | $F_{0,1,2,0}$ | $0$ | $F_{1,0,2,0}$ | $2^{2}\times 2^{4}.2^{3}$ | $(2^{5}\mathpunct{:}2^{2\cdot 3})\mathpunct{:}(\operatorname{Sym}_{3}\times\operatorname{Sym}_{3})$
$(2^{6})_{g}$ | $2{\rm BC}_{15}4{\rm A}_{40}4{\rm H}_{8}$ | $F_{1,0,1,1}$ | $2$ | $F_{0,0,1,1}$ | $2^{4}\times(T_{2}.2^{2})$ | $(2^{5}\mathpunct{:}\times 2^{4})\mathpunct{:}(2^{2}\mathpunct{:}\operatorname{Sym}_{3})$
$(2^{6})_{h}$ | $2{\rm BC}_{31}4{\rm A}_{32}$ | $F_{1,2}^{\prime\prime}$ | $1$ | $F_{0,2}^{\prime\prime}$ | $2^{6}\times(T_{1}.i)$ | $(2^{5}\mathpunct{:}2^{4})\mathpunct{:}{\rm Sp}_{4}(2)$
$(2^{5})_{a}$ | $2{\rm BC}_{7}4{\rm A}_{18}4{\rm H}_{6}$ | $F_{1,2}$ | $6$ | $F_{1,1}$ | $2^{2}\times(T_{3}\circ_{2}\operatorname{SL}_{2}(K)).(i{:}1)$ | $(2^{2}\times 2^{2})\mathpunct{:}(\operatorname{Sym}_{3}\times\operatorname{Sym}_{3})$
$(2^{5})_{b}$ | $2{\rm BC}_{7}4{\rm A}_{21}4{\rm H}_{3}$ | $F_{0,3}$ | $3$ | $F_{0,2}$ | $2^{5}\times{\rm PGL}_{2}(K)$ | $\operatorname{Sym}_{3}\times\operatorname{SL}_{3}(2)$
$(2^{5})_{c}$ | $2{\rm BC}_{7}4{\rm A}_{12}4{\rm H}_{12}$ | $F_{2,1}$ | $12$ | $F_{2,0}$ | $2^{2}\times(\operatorname{Spin}_{4}(K)\circ_{2}\operatorname{Spin}_{4}(K))$ | $(2^{2\cdot 2}\times 2^{2})\mathpunct{:}(\operatorname{Sym}_{3}\times\operatorname{Sym}_{3})$
$(2^{5})_{d}$ | $2{\rm BC}_{15}4{\rm A}_{14}4{\rm H}_{2}$ | $F_{1,3}^{\prime}$ | $5$ | $F_{0,3}^{\prime}$ | $2^{3}\times(T_{2}\circ_{2}\operatorname{SL}_{2}(K)).\gamma$ | $(2\times 2^{3})\mathpunct{:}(\operatorname{SL}_{3}(2))$
$(2^{5})_{e}$ | $2{\rm BC}_{7}4{\rm A}_{18}4{\rm H}_{6}$ | $F_{0,0,1,1}$ | $6$ | $F_{0,0,0,1}$ | $2^{4}\times(\operatorname{SL}_{2}(K)\circ_{2}\operatorname{SL}_{2}(K)).\leftrightarrow$ | $(2^{4}\mathpunct{:}2^{3})\mathpunct{:}\operatorname{Sym}_{3}$
$(2^{5})_{f}$ | $2{\rm BC}_{7}4{\rm A}_{20}4{\rm H}_{4}$ | $F_{1,0,2,0}$ | $4$ | $F_{2}^{\prime}$ | $2^{2}\times T_{4}.2^{3}$ | $(2^{4}\mathpunct{:}2^{2\cdot 2})\mathpunct{:}\operatorname{Sym}_{3}$
$(2^{5})_{g}$ | $2{\rm BC}_{7}4{\rm A}_{20}4{\rm H}_{4}$ | $F_{1,0,0,1}$ | $4$ | $F_{0,0,0,1}$ | $2^{4}\times(\operatorname{GL}_{2}(K)/2).\gamma$ | $(2^{4}\mathpunct{:}2^{2})\mathpunct{:}\operatorname{Sym}_{3}$
$(2^{5})_{h}$ | $2{\rm BC}_{7}4{\rm A}_{22}4{\rm H}_{2}$ | $F_{0,1,1,0}$ | $2$ | $F_{0,1,0,0}$ | $2^{4}\times T_{2}.D_{8}$ | $(2^{4}\mathpunct{:}2^{3})\mathpunct{:}\operatorname{Sym}_{3}$
$(2^{5})_{i}$ | $2{\rm BC}_{15}4{\rm A}_{16}$ | $F_{2,1}^{\prime\prime}$ | $3$ | $F_{1,1}^{\prime\prime}$ | $2^{3}\times T_{3}.2^{3}$ | $(2^{4}\mathpunct{:}2^{2\cdot 2})\mathpunct{:}(\operatorname{Sym}_{3}\times\operatorname{Sym}_{3})$
$(2^{5})_{j}$ | $2{\rm BC}_{15}4{\rm A}_{16}$ | $F_{0,2}^{\prime\prime}$ | $3$ | $F_{1,1}^{\prime\prime}$ | $2^{6}\times\operatorname{PSL}_{2}(K)$ | $2^{4}\mathpunct{:}{\rm Sp}_{4}(2)$
$(2^{5})_{k}$ | $2{\rm BC}_{7}4{\rm A}_{24}$ | $F_{3}^{\prime}$ | $0$ | $F_{2}^{\prime}$ | $2^{2}\times 2^{3+6}$ | $2^{3\cdot 2}\mathpunct{:}(\operatorname{SL}_{3}(2)\times\operatorname{Sym}_{3})$
$(2^{5})_{l}$ | $2{\rm BC}_{31}$ | $F_{2}^{\prime\prime}$ | $9$ | $F_{1}^{\prime\prime}$ | $2^{3}\times\operatorname{SL}_{2}(K)\circ_{2}(\operatorname{SL}_{2}(K))^{2}$ | $2^{2\cdot 3}\mathpunct{:}(\operatorname{Sym}_{3}\times\operatorname{SL}_{3}(2))$
$(2^{4})_{a}$ | $2{\rm BC}_{3}4{\rm A}_{9}4{\rm H}_{3}$ | $F_{0,2}$ | $10$ | $F_{0,1}$ | $2^{2}\times(T_{2}\circ_{3}\operatorname{SL}_{3}(K)).(i{:}\gamma)$ | $\operatorname{Sym}_{3}\times\operatorname{Sym}_{3}$
$(2^{4})_{b}$ | $2{\rm BC}_{3}4{\rm A}_{6}4{\rm H}_{6}$ | $F_{1,1}$ | $16$ | $F_{0,1}$ | $2^{2}\times(\operatorname{SL}_{2}(K)^{2}\circ_{2}{\rm Sp}_{4}(K))$ | $(2^{2}\times 2)\mathpunct{:}\operatorname{Sym}_{3}$
$(2^{4})_{c}$ | $2{\rm BC}_{3}4{\rm H}_{12}$ | $F_{2,0}$ | $28$ | $F_{1,0}$ | $2^{2}\times\operatorname{Spin}_{8}(K)$ | $2^{2\cdot 2}\mathpunct{:}(\operatorname{Sym}_{3}\times\operatorname{Sym}_{3})$
$(2^{4})_{d}$ | $2{\rm BC}_{7}4{\rm A}_{7}4{\rm H}_{1}$ | $F_{0,3}^{\prime}$ | $9$ | $F_{0}^{\prime\prime}$ | $2^{3}\times(\operatorname{GL}_{3}(K)/2).\gamma$ | $\operatorname{SL}_{3}(2)$
$(2^{4})_{e}$ | $2{\rm BC}_{3}4{\rm A}_{8}4{\rm H}_{4}$ | $F_{0,0,2,0}$ | $12$ | $F_{0,0,1,0}$ | $2^{2}\times((\operatorname{SL}_{2}(K))^{2}\circ_{2}(\operatorname{SL}_{2}(K))^{2}).2^{2}$ | $(2^{3}\mathpunct{:}2^{2})\mathpunct{:}\operatorname{Sym}_{3}$
$(2^{4})_{f}$ | $2{\rm BC}_{3}4{\rm A}_{9}4{\rm H}_{3}$ | $F_{0,0,0,1}$ | $10$ | $F_{0,1}^{\prime\prime}$ | $2^{4}\times\operatorname{PSp}_{4}(K)$ | $2^{3}\mathpunct{:}\operatorname{Sym}_{3}$
$(2^{4})_{g}$ | $2{\rm BC}_{3}4{\rm A}_{11}4{\rm H}_{1}$ | $F_{0,1,0,0}$ | $6$ | $F_{0,1}^{\prime\prime}$ | $2^{4}\times(\operatorname{PSL}_{2}(K))^{2}.\leftrightarrow$ | $2^{3}\mathpunct{:}\operatorname{Sym}_{3}$
$(2^{4})_{h}$ | $2{\rm BC}_{3}4{\rm A}_{10}4{\rm H}_{2}$ | $F_{1,0,1,0}$ | $8$ | $F_{1,0,0,0}$ | $2^{2}\times(\operatorname{GL}_{2}(K)\circ_{2}\operatorname{GL}_{2}(K)).2^{2}$ | $2^{3}\mathpunct{:}2^{2}$
$(2^{4})_{i}$ | $2{\rm BC}_{7}4{\rm A}_{8}$ | $F_{1,1}^{\prime\prime}$ | $7$ | $F_{0,1}^{\prime\prime}$ | $2^{3}\times((T_{1}.i)\circ_{2}\operatorname{SO}_{4}(K)).(i{:}\gamma)$ | $(2^{3}\mathpunct{:}2^{2})\mathpunct{:}\operatorname{Sym}_{3}$
$(2^{4})_{j}$ | $2{\rm BC}_{3}4{\rm A}_{12}$ | $F_{2}^{\prime}$ | $4$ | $F_{1}^{\prime}$ | $2^{2}\times T_{4}.2_{+}^{1+4}$ | $2^{2\cdot 2}\mathpunct{:}(\operatorname{Sym}_{3}\times\operatorname{Sym}_{3})$
$(2^{4})_{k}$ | $2{\rm BC}_{15}$ | $F_{1}^{\prime\prime}$ | $13$ | $F_{0}^{\prime\prime}$ | $2^{3}\times\operatorname{SL}_{2}(K)\circ_{2}{\rm Sp}_{4}(K)$ | $2^{3}\mathpunct{:}\operatorname{SL}_{3}(2)$
$(2^{3})_{a}$ | $2{\rm BC}_{1}4{\rm A}_{3}4{\rm H}_{3}$ | $F_{0,1}$ | $24$ | $F_{0,0,0,0}$ | $2^{2}\times\operatorname{SL}_{2}(K)\circ_{2}{\rm Sp}_{6}(K)$ | $\operatorname{Sym}_{3}$
$(2^{3})_{b}$ | $2{\rm BC}_{1}4{\rm H}_{6}$ | $F_{1,0}$ | $36$ | $F_{0,0}$ | $2^{2}\times\operatorname{Spin}_{9}(K)$ | $2^{2}\mathpunct{:}\operatorname{Sym}_{3}$
$(2^{3})_{c}$ | $2{\rm BC}_{1}4{\rm A}_{5}4{\rm H}_{1}$ | $F_{1,0,0,0}$ | $16$ | $F_{0,0,0,0}$ | $2^{2}\times(\operatorname{GL}_{4}(K)/Z).\gamma$ | $2^{2}$
$(2^{3})_{d}$ | $2{\rm BC}_{1}4{\rm A}_{4}4{\rm H}_{2}$ | $F_{0,0,1,0}$ | $20$ | $F_{0,0,0,0}$ | $2^{2}\times({\rm Sp}_{4}(K)\circ_{2}{\rm Sp}_{4}(K)).\leftrightarrow$ | $2^{2}\mathpunct{:}2$
$(2^{3})_{e}$ | $2{\rm BC}_{3}4{\rm A}_{4}$ | $F_{0,1}^{\prime\prime}$ | $15$ |  | $2^{3}\times\operatorname{PSO}_{6}(K).\gamma$ | $2^{2}\mathpunct{:}\operatorname{Sym}_{3}$
$(2^{3})_{f}$ | $2{\rm BC}_{1}4{\rm A}_{6}$ | $F_{1}^{\prime}$ | $12$ | $F_{0}^{\prime}$ | $2^{2}\times(\operatorname{SO}_{4}(K)\circ_{2}\operatorname{SO}_{4}(K)).\langle\gamma{:}\gamma,\leftrightarrow\rangle$ | $2^{2}\mathpunct{:}\operatorname{Sym}_{3}$
$(2^{3})_{g}$ | $2{\rm BC}_{7}$ | $F_{0}^{\prime\prime}$ | $21$ |  | $2^{3}\times\text{PSp}_{6}(K)$ | $\operatorname{SL}_{3}(2)$
$(2^{2})_{a}$ | $4{\rm H}_{3}$ | $F_{0,0}$ | $52$ |  | $2^{2}\times F_{4}(K)$ | $\operatorname{Sym}_{3}$
$(2^{2})_{b}$ | $4{\rm A}_{2}{\rm H}_{1}$ | $F_{0,0,0,0}$ | $36$ |  | $2^{2}\times\operatorname{PSp}_{8}(K)$ | $2$
$(2^{2})_{c}$ | $4{\rm A}_{3}$ | $F_{0}^{\prime}$ | $28$ |  | $2^{2}\times\operatorname{P\Omega}_{8}(K)$ | $\operatorname{Sym}_{3}$
Table 7. Non-toral elementary abelian $2$-subgroups of the algebraic group $G=E_{8}$; here $L=E_{7,{\text{\rm ad}}}\leqslant G$ and the subgroup given in the column labelled “$X\leqslant L$” is from Table 6 and used in Section 6.8. $\boldsymbol{E}$ | $\boldsymbol{\rm{Dist}(E)}$ | Name | $\dim$ | $X\leqslant L$ | $\boldsymbol{C_{G}(E)}$ | $\boldsymbol{N_{G}(E)/C_{G}(E)}$
---|---|---|---|---|---|---
$2^{9}$ | $2{\rm A}_{120}{\rm B}_{391}$ | $F_{0,1,0,2}$ | $0$ | $F_{0,1,0,2}$ | $2^{9}$ | $2^{8}\mathpunct{:}\operatorname{SO}_{8}^{+}(2)$
$(2^{8})_{a}$ | $2{\rm A}_{56}{\rm B}_{199}$ | $F_{2,3}$ | $0$ | $F_{2,3}$ | $2^{8}$ | $2^{2\cdot 6}\mathpunct{:}(\operatorname{Sym}_{3}\times(\operatorname{SL}_{3}(2)\wr 2))$
$(2^{8})_{b}$ | $2{\rm A}_{56}{\rm B}_{199}$ | $F_{0,1,1,1}$ | $0$ | $F_{0,1,1,1}$ | $2^{9}$ | $(2\times 2^{6})\mathpunct{:}(2^{6}\mathpunct{:}(\operatorname{SO}_{6}^{+}(2)))$
$(2^{8})_{c}$ | $2{\rm A}_{64}{\rm B}_{191}$ | $F_{1,0,0,2}$ | $1$ | $F_{1,0,0,2}$ | $2^{7}\times T_{1}.2$ | $2^{7}\mathpunct{:}\operatorname{SO}_{7}(2)$
$(2^{7})_{a}$ | $2{\rm A}_{40}{\rm B}_{87}$ | $F_{2,2}$ | $4$ | $F_{2,2}$ | $2^{3}\times T_{4}.2$ | $(2^{2\cdot 2\cdot 1}\times 2^{2\cdot 1\cdot 3})\mathpunct{:}(\operatorname{Sym}_{3}\times\operatorname{Sym}_{3}\times\operatorname{SL}_{3}(2))$
$(2^{7})_{b}$ | $2{\rm A}_{28}{\rm B}_{99}$ | $F_{1,3}$ | $1$ | $F_{1,3}$ | $2^{6}\times T_{1}.2$ | $2^{6}\mathpunct{:}(\operatorname{SL}_{3}(2)\wr 2)$
$(2^{7})_{c}$ | $2{\rm A}_{24}{\rm B}_{103}$ | $F_{0,1,2,0}$ | $0$ | $F_{0,1,2,0}$ | $2^{3}\times 2^{4}.2^{3}$ | $(2^{2}\times 2^{4})\mathpunct{:}(2^{2\cdot 4}\mathpunct{:}(\operatorname{Sym}_{3}\times\operatorname{SO}_{4}^{+}(2)))$
$(2^{7})_{d}$ | $2{\rm A}_{28}{\rm B}_{99}$ | $F_{0,1,0,1}$ | $1$ | $F_{0,1,0,1}$ | $2^{7}\times T_{1}.2$ | $2^{6}\mathpunct{:}\operatorname{SO}_{6}^{+}(2)$
$(2^{7})_{e}$ | $2{\rm A}_{32}{\rm B}_{95}$ | $F_{1,0,1,1}$ | $2$ | $F_{1,0,1,1}$ | $2^{5}\times T_{2}.2^{2}$ | $(2\times 2^{5})\mathpunct{:}(2^{5}\mathpunct{:}\operatorname{SO}_{5}(2))$
$(2^{7})_{f}$ | $2{\rm A}_{36}{\rm B}_{91}$ | $F_{0,0,0,2}$ | $3$ | $F_{0,0,0,2}$ | $2^{7}\times{\rm PSL}_{2}(K)$ | $2^{6}\mathpunct{:}\operatorname{SO}_{6}^{-}(2)$
$(2^{6})_{a}$ | $2{\rm A}_{32}{\rm B}_{31}$ | $F_{2,1}$ | $12$ | $F_{2,1}$ | ${2^{3}}\times(\operatorname{SL}_{2}(K)^{2})\circ_{2}(\operatorname{SL}_{2}(K)^{2})$ | $(2^{2\cdot 1}\times 2^{2\cdot 3})\mathpunct{:}(\operatorname{Sym}_{3}\times\operatorname{SL}_{3}(2))$
$(2^{6})_{b}$ | $2{\rm A}_{20}{\rm B}_{43}$ | $F_{1,2}$ | $6$ | $F_{1,2}$ | $2^{3}\times(T_{3}\circ_{2}\operatorname{SL}_{2}(K)).2$ | $(2^{2\cdot 1}\times 2^{1\cdot 3})\mathpunct{:}(\operatorname{Sym}_{3}\times\operatorname{SL}_{3}(2))$
$(2^{6})_{c}$ | $2{\rm A}_{14}{\rm B}_{49}$ | $F_{0,3}$ | $3$ | $F_{0,3}$ | $2^{6}\times{\rm PGL}_{2}(K)$ | $\operatorname{SL}_{3}(2)\wr 2$
$(2^{6})_{d}$ | $2{\rm A}_{12}{\rm B}_{51}$ | $F_{0,1,1,0}$ | $2$ | $F_{0,1,1,0}$ | $2^{5}\times T_{2}.D_{8}$ | $(2\times 2^{4})\mathpunct{:}(2^{4}\mathpunct{:}(\operatorname{SO}_{4}^{+}(2)))$
$(2^{6})_{e}$ | $2{\rm A}_{16}{\rm B}_{47}$ | $F_{1,0,2,0}$ | $4$ | $F_{1,0,2,0}$ | $2^{3}\times T_{4}.2^{3}$ | $(2^{2}\times 2^{3})\mathpunct{:}(2^{2\cdot 3}\mathpunct{:}(\operatorname{Sym}_{3}\times\operatorname{SO}_{3}(2)))$
$(2^{6})_{f}$ | $2{\rm A}_{16}{\rm B}_{47}$ | $F_{1,0,0,1}$ | $4$ | $F_{1,0,0,1}$ | $2^{5}\times({\rm GL}_{2}(K)/2).2$ | $2^{5}\mathpunct{:}\operatorname{SO}_{5}(2)$
$(2^{6})_{g}$ | $2{\rm A}_{20}{\rm B}_{43}$ | $F_{0,0,1,1}$ | $6$ | $F_{0,0,1,1}$ | $2^{5}\times(\operatorname{SL}_{2}(K)\circ_{2}\operatorname{SL}_{2}(K)).2$ | $(2\times 2^{4})\mathpunct{:}(2^{4}\mathpunct{:}(\operatorname{SO}_{4}^{-}(2)))$
$(2^{6})_{h}$ | $2{\rm A}_{8}{\rm B}_{55}$ | $F_{3,2}^{\prime\prime}$ | $0$ | $F_{3}^{\prime}$ | $2^{3}\times 2^{3+6}$ | $2^{3\cdot 2}\mathpunct{:}((2^{3}\times 2^{2})\mathpunct{:}(\operatorname{SL}_{3}(2)\times\operatorname{Sym}_{3}))$
$(2^{5})_{a}$ | $2{\rm A}_{28}{\rm B}_{3}$ | $F_{2,0}$ | $28$ | $F_{2,0}$ | $2^{3}\times\operatorname{Spin}_{8}(K)$ | $2^{2\cdot 3}\mathpunct{:}(\operatorname{Sym}_{3}\times\operatorname{SL}_{3}(2))$
$(2^{5})_{b}$ | $2{\rm A}_{16}{\rm B}_{15}$ | $F_{1,1}$ | $16$ | $F_{1,1}$ | $2^{3}\times(\operatorname{SL}_{2}(K)^{2}\circ_{2}{\rm Sp}_{4}(K))$ | $2\times 2^{3}\mathpunct{:}\operatorname{SL}_{3}(2)$
$(2^{5})_{c}$ | $2{\rm A}_{10}{\rm B}_{21}$ | $F_{0,2}$ | $10$ | $F_{0,2}$ | $2^{3}\times(T_{1}\times\operatorname{GL}_{3}(K)).2$ | $\operatorname{Sym}_{3}\times\operatorname{SL}_{3}(2)$
$(2^{5})_{d}$ | $2{\rm A}_{6}{\rm B}_{25}$ | $F_{0,1,0,0}$ | $6$ | $F_{0,1,0,0}$ | $2^{5}\times{{\rm PSL}_{2}(K)^{2}}.2$ | $2^{4}\mathpunct{:}\operatorname{SO}_{4}^{+}(2)$
$(2^{5})_{e}$ | $2{\rm A}_{8}{\rm B}_{23}$ | $F_{1,0,1,0}$ | $8$ | $F_{1,0,1,0}$ | $2^{3}\times(T_{2}\circ_{2^{2}}(\operatorname{SL}_{2}(K)^{2})).2^{2}$ | $(2\times 2^{3})\mathpunct{:}(2^{3}\mathpunct{:}(\operatorname{SO}_{3}(2)))$
$(2^{5})_{f}$ | $2{\rm A}_{12}{\rm B}_{19}$ | $F_{0,0,2,0}$ | $12$ | $F_{0,0,2,0}$ | $2^{3}\times((\operatorname{SL}_{2}(K)^{2})\circ_{2}(\operatorname{SL}_{2}(K)^{2})).2^{2}$ | $(2^{2}\times 2^{2})\mathpunct{:}(2^{2\cdot 2}\mathpunct{:}(\operatorname{Sym}_{3}\times\operatorname{Sym}_{3}))$
$(2^{5})_{g}$ | $2{\rm A}_{10}{\rm B}_{21}$ | $F_{0,0,0,1}$ | $10$ | $F_{0,0,0,1}$ | $2^{5}\times{\rm PSp}_{4}(K)$ | $2^{4}\mathpunct{:}(\operatorname{SO}_{4}^{-}(2))$
$(2^{5})_{h}$ | $2{\rm A}_{4}{\rm B}_{27}$ | $F_{2,2}^{\prime\prime}$ | $4$ | $F_{2}^{\prime}$ | ${2^{3}}\times T_{4}.2_{+}^{1+4}$ | $2^{2\cdot 2}\mathpunct{:}((2^{2}\times 2^{2})\mathpunct{:}(\operatorname{Sym}_{3}\times\operatorname{Sym}_{3}))$
$(2^{5})_{i}$ | $2{\rm B}_{31}$ | $F_{5}^{\prime}$ | $0$ |  | $2^{5+10}$ | $\operatorname{SL}_{5}(2)$
$(2^{4})_{a}$ | $2{\rm A}_{14}{\rm B}_{1}$ | $F_{1,0}$ | $36$ | $F_{1,0}$ | $2^{3}\times\operatorname{Spin}_{9}(K)$ | $2^{3}\mathpunct{:}\operatorname{SL}_{3}(2)$
$(2^{4})_{b}$ | $2{\rm A}_{8}{\rm B}_{7}$ | $F_{0,1}$ | $24$ | $F_{0,1}$ | $2^{3}\times\operatorname{SL}_{2}(K)\circ_{2}{\rm Sp}_{6}(K)$ | $\operatorname{SL}_{3}(2)$
$(2^{4})_{c}$ | $2{\rm A}_{4}{\rm B}_{11}$ | $F_{1,0,0,0}$ | $16$ | $F_{1,0,0,0}$ | $2^{3}\times(\operatorname{GL}_{4}(K)/2).2$ | $2^{3}\mathpunct{:}\operatorname{SO}_{3}(2)$
$(2^{4})_{d}$ | $2{\rm A}_{6}{\rm B}_{9}$ | $F_{0,0,1,0}$ | $20$ | $F_{0,0,1,0}$ | $2^{3}\times({\rm Sp}_{4}(K)\circ_{2}{\rm Sp}_{4}(K)).2$ | $2^{3}\mathpunct{:}(2^{2}\mathpunct{:}\operatorname{Sym}_{3})$
$(2^{4})_{e}$ | $2{\rm A}_{2}{\rm B}_{13}$ | $F_{1,2}^{\prime\prime}$ | $12$ | $F_{1}^{\prime}$ | $2^{3}\times(\operatorname{SO}_{4}(K)\circ_{2}\operatorname{SO}_{4}(K)).2^{2}$ | $2^{1\cdot 2}\mathpunct{:}((2\times 2^{2})\mathpunct{:}\operatorname{Sym}_{3})$
$(2^{3})_{a}$ | $2{\rm A}_{7}$ | $F_{0,0}$ | $52$ | $F_{0,0}$ | $2^{3}\times F_{4}(K)$ | $\operatorname{SL}_{3}(2)$
$(2^{3})_{b}$ | $2{\rm A}_{3}{\rm B}_{4}$ | $F_{0,0,0,0}$ | $36$ | $F_{0,0,0,0}$ | $2^{3}\times{\rm P}{\rm Sp}_{8}(K)$ | $2^{2}\mathpunct{:}\operatorname{Sym}_{3}$
$(2^{3})_{c}$ | $2{\rm A}_{1}{\rm B}_{6}$ | $F_{0,2}^{\prime\prime}$ | $28$ | $F_{0}^{\prime}$ | $2^{3}\times{\rm P}\Omega_{8}(K)$ | $2^{2}\mathpunct{:}\operatorname{Sym}_{3}$

It follows from our description in Section 5.1 that the number of toral
subgroups does not depend on the isogeny type (although the class
distributions and normaliser structure of the subgroups do vary). On the other
hand, [40, Theorem 1.1] tells us that for a group $G$ of type $G_{2}$,
$F_{4}$, $E_{6}$, $E_{7}$, and $E_{8}$ we respectively have exactly $4$, $12$,
$51$, $78$, and $66$ classes of elementary abelian $2$-subgroups in
$\operatorname{Aut}(\operatorname{Lie}(G))$. If $G$ is adjoint then
$G=\operatorname{Aut}(\operatorname{Lie}(G))$, unless $G$ has type $E_{6}$ in
which case by inspecting [40, §3.4] we see that
$G=\operatorname{Aut}(\operatorname{Lie}(G))^{\circ}$ contains $21$ classes of
elementary abelian $2$-subgroups (using the notation of [40], these are the
classes in ‘Class 3’ and ‘Class 4’).

Thus for adjoint groups of type $G_{2}$, $F_{4}$, $E_{6}$, $E_{7}$, and
$E_{8}$ there are respectively $1$, $3$, $3$, $46$, and $35$ classes of non-
toral elementary abelian $2$-subgroups. So if we verify the information stated
in the tables, it becomes evident that all the listed subgroups are non-
conjugate, and counting then shows that these are all non-toral subgroups with
elementary abelian image in $G/Z(G)$. For the simply connected group
$E_{7,{\text{\rm sc}}}$, which is not isomorphic to its adjoint version as the
characteristic is not $2$, we provide our own detailed calculations, showing
explicitly that the $11$ given classes of subgroups above constitute all non-
toral subgroups.

We now split the proof into several cases.

###  6.3. Proof of Proposition 6.3 for $G=G_{2}(K)$.

This follows directly from [22, Table I].

###  6.4. Proof of Proposition 6.3 for $G=F_{4}(K)$.

The collection of all elementary abelian $2$-subgroups of $G$ is described in
[40, Proposition 5.2]. These twelve subgroups are labelled $F_{r,s}$ for
$r\leqslant 2$ and $s\leqslant 3$, and it follows directly from the proof of
[loc. cit.] that the class distribution of $F_{r,s}$ is $2{\rm
A}_{2^{r+s}-2^{r}-2}{\rm B}_{2^{r}-1}$. Using our algorithm from Section 5.1,
we find that $9$ of these $12$ possible class distributions are the
distribution of a toral elementary abelian $2$-subgroup; the exceptions are
$2{\rm A}_{28}{\rm B}_{3}=\operatorname{Dist}(F_{2,3})$, $2{\rm A}_{14}{\rm
B}_{1}=\operatorname{Dist}(F_{1,3})$, and $2{\rm
A}_{7}=\operatorname{Dist}(F_{0,3})$. We conclude that $G$ has exactly three
conjugacy classes of non-toral subgroups (cf. also [22, Table I and Theorem
7.3]). The structure of $N_{G}(F_{r,s})/C_{G}(F_{r,s})$ for each subgroup
occurring is given in [40, Proposition 5.5]; it remains to consider their
centralisers. Using Lemma 3.5, the subgroups of order $2^{3}$, $2^{4}$, and
$2^{5}$ have centralisers of dimension $3$, $1$, and $0$, respectively. By
[31, Theorem 1], the group $G$ has a maximal connected subgroup of type
$A_{1}G_{2}$, where the factor $A_{1}$ is adjoint as this maximal subgroup is
not contained in an involution centraliser in $G$. The non-toral subgroup
$E=2^{3}$ of this $G_{2}$ factor, having all involutions conjugate, must be
non-toral in $G$. Thus $E$ is a representative of the $G$-class of non-toral
subgroups $2^{3}$. Now $E$ centralises a subgroup $A_{1}$, and we deduce that
$C_{G}(E)^{\circ}\cong\operatorname{PGL}_{2}(K)$. Moreover, $C_{G}(E)$
normalises $C_{G}(E)^{\circ}$; since the latter has no outer automorphisms,
$C_{G}(E)\leqslant\operatorname{PGL}_{2}(K)C_{G}(\operatorname{PGL}_{2}(K))$.
From the maximality of the subgroup of type $A_{1}G_{2}$ it follows that
$C_{G}(\operatorname{PGL}_{2}(K))=G_{2}(K)$, hence
$C_{G}(E)\leqslant\operatorname{PGL}_{2}(K)G_{2}(K)$. Since $E<G_{2}(K)$ is
the unique non-toral subgroup and $C_{G_{2}(K)}(E)=E$, we have
$C_{G}(E)=E\times\operatorname{PGL}_{2}(K)$. If $E$ is non-toral of order
$2^{4}$, then $C_{G}(E)$ is the centraliser of a non-central involution of
$2^{3}\times\operatorname{PGL}_{2}(K)$, and is therefore
$2^{3}\times(T_{1}\mathpunct{:}2)$. Finally, when $E$ is non-toral of order
$2^{5}$ we have $E\leqslant C_{G}(E)\leqslant
C_{2^{3}\times(T_{1}\mathpunct{:}2)}(E)=2^{3}\times(2^{2})=E$.

###  6.5. Proof of Proposition 6.3 for $G=E_{6}(K)$.

Recall that $Z$ is the kernel of the isogeny from the simply connected cover
of $G$. From [22, Table I] it follows that, up to conjugacy, the three non-
toral subgroups of the simply connected group of type $E_{6}$ are precisely
the images of the non-toral subgroups of $F_{4}(K)$ under the inclusion
$F_{4}(K)\to E_{6,{\text{\rm sc}}}(K)$: in the terminology of [22], this is
because these are each precisely the subgroups of complexity at least 3. Their
class distributions in Table 5 follow from their distributions in $F_{4}(K)$.
Using Lemma 3.5, the centralisers of $E=2^{3}$, $2^{4}$, and $2^{5}$ have
dimension $8$, $4$, and $2$, respectively. By [22, Remark 8.3], the group $G$
has a maximal subgroup $M$ of type $A_{2}G_{2}$. The unique non-toral
$2^{3}\leqslant G_{2}$ is non-toral in $G$ (since all its involutions are
conjugate), and extends to a non-toral $2^{5}\leqslant G$ contained in $M$.
The known dimension tells us that $C_{G}(2^{3})^{\circ}$ is precisely the
factor of type $A_{2}$. We claim that its centre is $Z(G_{{\text{\rm
sc}}})/Z$. It follows from [31, Tables 10.1 & 10.2] that this factor $A_{2}$
has a natural $3$-dimensional module (high weight ‘$10$’) as a composition
factor of the $27$-dimensional module, hence this factor $A_{2}$ has a non-
trivial centre when $G$ is simply connected; on the other hand, all
composition factors of this subgroup $A_{2}$ on $L(G)$ have high weight ‘$11$’
or ‘$00$’, hence occur as sub-quotients of $10\otimes 01$. Thus the centre of
this subgroup $A_{2}$ acts trivially on $L(G)$, and so lies in $\ker(G\to
G_{{\text{\rm ad}}})=Z(G_{{\text{\rm sc}}})/Z$. Since $M$ is maximal, the
normaliser of $C_{G}(2^{3})^{\circ}$ in $G$ is $M$, hence
$C_{G}(2^{3})\leqslant M$. Since $M$ is a direct product and $2^{3}<G_{2}$, we
have

$C_{G}(2^{3})=\operatorname{SL}_{3}(K)/Z\times
C_{G_{2}(K)}(2^{3})=2^{3}\times\operatorname{SL}_{3}(K)/Z.$

Since $C_{G}(2^{5})\leqslant C_{G}(2^{4})\leqslant
C_{G}(2^{3})=2^{3}\times\operatorname{SL}_{3}(K)/Z$, we observe that
$C_{G}(2^{4})^{\circ}$ is a $4$-dimensional reductive subgroup of
$\operatorname{SL}_{3}(K)/Z$, and is therefore a Levi subgroup isomorphic to
$\operatorname{GL}_{2}(K)$. The centraliser $C_{G}(2^{4})$ normalises this
$\operatorname{GL}_{2}(K)$, hence is equal to
$2^{3}\times\operatorname{GL}_{2}(K)$. Next, $C_{G}(2^{5})^{\circ}$ is
reductive and $2$-dimensional, hence is a subtorus of the subgroup
$\operatorname{GL}_{2}(K)$ above. The centraliser $C_{G}(2^{5})$ is contained
in $2^{3}\times\operatorname{GL}_{2}(K)$ and normalises this torus, hence is
the product of $2^{3}$ with an elementary abelian subgroup of
$\operatorname{GL}_{2}(K)$. Every such subgroup is toral in
$\operatorname{GL}_{2}(K)$ by [32, Corollary 14.17], hence
$C_{G}(2^{5})=2^{3}\times T_{2}$. Finally, the groups $N_{G}(E)/C_{G}(E)$ are
given in [40, Proposition 6.10], but we correct errors for $2^{5}$ and
$2^{4}$. Note that the three non-toral groups are $F_{0,3}$, $F_{1,3}$, and
$F_{2,3}$ as defined in [40, p. 272], see also Case $F_{4}$ above. Since
$2^{5}\leqslant F_{4}(K)\leqslant G$ and
$\operatorname{Out}_{F_{4}(K)}(2^{5})=2^{2\cdot
3}.(\operatorname{Sym}_{3}\times\operatorname{SL}_{3}(2))$, we have that
$\operatorname{Out}_{G}(2^{5})\geqslant 2^{2\cdot
3}.(\operatorname{Sym}_{3}\times\operatorname{SL}_{3}(2))$. Note that
$E=2^{5}$ contains involutions $2{\rm A}$ and $2{\rm B}$, so
$\operatorname{Out}_{G}(2^{5})$ is a proper subgroup of
$\operatorname{Out}(2^{5})=\operatorname{SL}_{5}(2)$. But $2^{2\cdot
3}.(\operatorname{Sym}_{3}\times\operatorname{SL}_{3}(2))$ is a maximal
(parabolic) subgroup of $\operatorname{Out}(2^{5})$, so
$\operatorname{Out}_{G}(2^{5})=\operatorname{Out}_{F_{4}(K)}(2^{5})$.
Similarly,
$N_{G}(2^{4})=(2^{3}\times\operatorname{GL}_{2}(K))(2^{3}\mathpunct{:}\operatorname{SL}_{3}(2))$,
since
$\operatorname{Out}_{F_{4}(K)}(2^{4})=2^{3}\mathpunct{:}\operatorname{SL}_{3}(2)\leqslant\operatorname{Out}(2^{4})=\operatorname{SL}_{4}(2)$
is maximal.

###  6.6. Proof of Proposition 6.3 for $G=E_{7,{{\text{\rm sc}}}}(K)$.

By [22, Table I] there exists a unique maximal non-toral subgroup $M=2^{6}$ of
$G$ with

$C_{G}(M)=2^{3}\times\operatorname{SL}_{2}(K)^{3}\quad\text{and}\quad
N_{G}(M)/C_{G}(M)=2^{2\cdot
3}.(\operatorname{SL}_{3}(2)\times\operatorname{Sym}_{3}).$

We now determine a set of generators for $M$ which allow us to easily
determine the class of an element of $M$, and to concretely realise
$N_{G}(M)/C_{G}(M)$ as the subgroup of $\operatorname{GL}_{6}(2)$ of all
matrices of the form

(1) $\displaystyle\begin{pmatrix}A&\mathbf{0}&\mathbf{0}\\\
\ast&B&\mathbf{0}\\\ \mathbf{0}&\mathbf{0}&1\end{pmatrix}$

where $A\in\operatorname{SL}_{3}(2)$, $B\in\operatorname{SL}_{2}(2)$, and
$\ast$ denotes an arbitrary $2\times 3$ matrix.

First, write $(2^{3})_{a}$ for the non-toral subgroup $2^{3}$ arising from
$E_{6}$. This remains non-toral in $G$ by [22, Remark 8.4], has distribution
$2{\rm B}_{7}$ in $G$, and we have shown above that it lies in the $G_{2}$
factor of a subgroup of type $A_{2}G_{2}$, maximal among connected subgroups
of $E_{6}(K)$. Since the centraliser in $G$ of the factor $G_{2}$ contains a
subgroup $A_{2}$, by inspecting [29, Table 8.2] it follows that the connected
centraliser of this $G_{2}$ factor is simple of type $C_{3}$. Using Lemma 3.5,
we deduce that $C_{G}((2^{3})_{a})$ has dimension $21$, and therefore
$C_{G}((2^{3})_{a})^{\circ}$ is simple of type $C_{3}$. Inspecting [29, Tables
8.2 & 8.6] also shows that the centre of this subgroup of type $C_{3}$ equals
$Z(G)=\langle z\rangle$ since it acts trivially on $L(G)$. Now the subgroup of
$G$ of type $G_{2}C_{3}$ is a maximal subgroup of $G$, and so each factor is
the full centraliser of the other. Hence $N_{G}((2^{3})_{a})\leqslant
N_{G}(C_{G}((2^{3})_{a})^{\circ})=G_{2}(K)\times{\rm Sp}_{6}(K)$. It follows
that

$N_{G}((2^{3})_{a})=(2^{3}\mathpunct{:}\operatorname{SL}_{3}(2))\times{\rm
Sp}_{6}(K)\quad\text{and}\quad C_{G}((2^{3})_{a})=2^{3}\times{\rm Sp}_{6}(K).$

The above also shows that $M\leqslant C_{G}((2^{3})_{a})\leqslant
G_{2}(K)\times{\rm Sp}_{6}(K)$. Since $(2^{3})_{a}\leqslant G_{2}$ and
$Z(C_{G}(M)^{\circ})=Z(\operatorname{SL}_{2}(K)^{3})\leqslant{\rm Sp}_{6}(K)$,
we obtain

$M=2^{6}=(2^{3})_{a}\times Z(\operatorname{SL}_{2}(K)^{3})=(M\cap
G_{2}(K))\times(M\cap{\rm Sp}_{6}(K)).$

We now use this decomposition to describe the $G$-classes of elements of $M$
and the action of $N_{G}(M)$. From [31, Table 10.2], the subgroup
$G_{2}(K)\times{\rm Sp}_{6}(K)$ acts on the $56$-dimensional $G$-module with
two composition factors; one is a tensor product $V_{7}\otimes V_{6}$ of the
natural $7$\- and $6$-dimensional modules for $G_{2}(K)$ and ${\rm
Sp}_{6}(K)$, denoted in [31] by their high weights ‘$10$’ and ‘$100$’. The
other is a $14$-dimensional irreducible ${\rm Sp}_{6}(K)$-module, $V_{14}$
(high weight ‘$001$’). Straightforward weight calculations show that the the
alternating third power of $V_{6}$ is isomorphic as an ${\rm
Sp}_{6}(K)$-module to $V_{6}\oplus V_{14}$. Hence the eigenvalues of an
element on $V_{14}$ can be determined directly from those on $V_{6}$. In
particular, letting $\chi_{7}$, $\chi_{6}$, and $\chi_{14}$ denote the
characters of the three respective modules, if $x\in(2^{3})_{a}$ and $y\in
Z(\operatorname{SL}_{2}(K)^{3})$, then $\chi_{\rm
min}(xy)=\chi_{7}(x)\chi_{6}(y)+\chi_{14}(y)$; see Section 3.3 for the
definition of $\chi_{\rm min}$. For $i=1,2,3$, let $y_{i}$ be a generator of
the centre of the $i$-th factor in $\operatorname{SL}_{2}(K)^{3}$, and write
$z=y_{1}y_{2}y_{3}\in Z(G)$. Each $y_{i}$ is ${\rm Sp}_{6}(K)$-conjugate to a
diagonal matrix $\operatorname{diag}(-1,-1,1,1,1,1)$, hence has trace $2$ on
$V_{6}$ and $-6$ on $V_{14}$. The products $y_{i}y_{j}$ $(i\neq j)$ have trace
$-2$ on $V_{6}$ and $6$ on $V_{14}$, and $y_{1}y_{2}y_{3}$ has trace $-6$ on
$V_{6}$ and $-14$ on $V_{14}$. Therefore if $x\in(2^{3})_{a}$ and $y\in
Z(\operatorname{SL}_{2}(K)^{3})$, we find

$\chi_{\rm min}(xy)=\left\\{\begin{array}[]{rll}56&\textup{:\ }x=1,\ y=1\\\
-56&\textup{:\ }x=1,\ y=y_{1}y_{2}y_{3}(=z)\\\ 8&\textup{:\ }x=1,\
y\in\\{y_{1},y_{2},y_{3}\\},&\textup{or }x\neq 1,\
y\in\\{1,y_{1}y_{2},y_{2}y_{3},y_{1}y_{3}\\}\\\ -8&\textup{:\ }x\neq 1,\
y\in\\{y_{1},y_{2},y_{3},z\\},&\textup{or }x=1,\
y\in\\{y_{1}y_{2},y_{2}y_{3},y_{1}y_{3}\\}\end{array}\right.$

which allows us to determine the distribution of $M$ via Table 1 as $2{\rm
A}_{1}{\rm B}_{31}{\rm C}_{31}$. Now, $N_{G}(M)$ acts on
$\operatorname{SL}_{2}(K)^{3}$, and hence on $\\{y_{1},y_{2},y_{3}\\}$, via
its subgroup $\operatorname{Sym}_{3}$. The subgroup
$(2^{3})_{a}\times\left<y_{1}y_{2},y_{2}y_{3}\right>$ is then the unique
subgroup of $M$ containing all $2{\rm B}$-involutions not lying in the
$N_{G}(M)$-orbit $\\{y_{1},y_{2},y_{3}\\}$, and is therefore
$N_{G}(M)$-invariant. Thus if $(2^{3})_{a}=\left<x_{1},x_{2},x_{3}\right>$
then $N_{G}(M)$ preserves the direct-product decomposition $M=2^{6}=\langle
x_{1},x_{2},x_{3},y_{1}y_{2},y_{2}y_{3}\rangle\times\left<z\right>$, and
taking the elements shown as an ordered vector-space basis for $M$, we realise
$N_{G}(M)/C_{G}(M)$ as the set of matrices as in (1) above.

Now define $(2^{3})_{b}=\left<x_{1},x_{2},zx_{3}\right>$, and note that
$C_{G}((2^{3})_{b})=C_{G}((2^{3})_{a})$, so $(2^{3})_{b}$ is non-toral by
Lemma 3.5 and has distribution $2{\rm B}_{3}{\rm C}_{4}$ by Table 1. Now a
direct calculation with (1) shows that $N_{G}(M)$ has eleven orbits on
subgroups of $M$ containing a conjugate of $(2^{3})_{a}$ or $(2^{3})_{b}$: in
addition to the subgroups $M$, $(2^{3})_{a}$, and $(2^{3})_{b}$, there are
subgroups

$\displaystyle(2^{4})_{a}$ $\displaystyle=2{\rm A}_{1}{\rm B}_{7}{\rm
C}_{7}=\left<x_{1},x_{2},x_{3},z\right>,$ $\displaystyle(2^{4})_{b}$
$\displaystyle=2{\rm B}_{14}{\rm
C}_{1}=\left<x_{1},x_{2},x_{3},zy_{1}\right>,$ $\displaystyle(2^{4})_{c}$
$\displaystyle=2{\rm B}_{8}{\rm C}_{7}=\left<x_{1},x_{2},x_{3},y_{1}\right>,$
$\displaystyle(2^{4})_{d}$ $\displaystyle=2{\rm B}_{6}{\rm
C}_{9}=\left<x_{1},x_{2},zx_{3},zy_{1}\right>,$ $\displaystyle(2^{5})_{a}$
$\displaystyle=2{\rm A}_{1}{\rm B}_{15}{\rm
C}_{15}=\left<x_{1},x_{2},x_{3},y_{1},z\right>,$ $\displaystyle(2^{5})_{b}$
$\displaystyle=2{\rm B}_{28}{\rm
C}_{3}=\left<x_{1},x_{2},x_{3},zy_{1},zy_{2}\right>,$
$\displaystyle(2^{5})_{c}$ $\displaystyle=2{\rm B}_{16}{\rm
C}_{15}=\left<x_{1},x_{2},x_{3},y_{1},y_{2}\right>,$
$\displaystyle(2^{5})_{d}$ $\displaystyle=2{\rm B}_{12}{\rm
C}_{19}=\left<x_{1},x_{2},zx_{3},zy_{1},zy_{2}\right>.$

Each such subgroup is non-toral, and their distributions, given in Table 5,
show that the subgroups are pairwise non-conjugate in $G$. There are $15$
further $N_{G}(M)$-orbits on subgroups of $M$ of rank $3$ or more, and each of
these has an $M$-conjugate contained in the subgroup
$\left<x_{1},x_{2},y_{1},y_{2},y_{3}\right>$. We claim that this subgroup, and
therefore all of its subgroups, are toral. By construction,
$\left<x_{1},x_{2},x_{3}\right>$ is contained in a subgroup $G_{2}$ and
$\left<y_{1},y_{2},y_{3}\right>$ is a central subgroup of
$\operatorname{SL}_{2}(K)^{3}$, which is in turn contained in a simple
subgroup of type $C_{3}$, and these subgroups $G_{2}$ and $C_{3}$ commute.
Moreover, $\left<x_{1},x_{2}\right>$ is toral in $G_{2}(K)$, and
$\left<y_{1},y_{2},y_{3}\right>$ is toral in $\operatorname{SL}_{2}(K)^{3}$ by
Lemma 3.5. The claim follows.

Centralisers. We now determine centralisers. Each subgroup above is generated
by either $(2^{3})_{a}$ or $(2^{3})_{b}$ together with some involutions in
${\rm Sp}_{6}(K)$. The ${\rm Sp}_{6}(K)$-centraliser of a non-central
involution is $\operatorname{SL}_{2}(K)\times{\rm Sp}_{4}(K)$, and it is then
straightforward to see that the ${\rm Sp}_{6}(K)$-centraliser of an elementary
abelian $2$-subgroup of ${\rm Sp}_{6}(K)$ is either ${\rm Sp}_{6}(K)$ itself,
or $\operatorname{SL}_{2}(K)\times{\rm Sp}_{4}(K)$, or
$\operatorname{SL}_{2}(K)^{3}$. This determines the centraliser of every non-
toral subgroup of $M$.

Normalisers. It remains to determine the normaliser quotient
$N_{G}(E)/C_{G}(E)$ for each subgroup $E$. Firstly, the non-toral subgroups of
$E_{6}$ remain non-toral in $G$ under the inclusion $3.E_{6}(K)\to G$ as they
each contain $(2^{3})_{a}$. These three subgroups are each contained in a
subgroup $F_{4}(K)$ by construction, and then [40, Propositions 5.4 & 5.5]
show that every possible class-preserving automorphism of such a subgroup is
already induced by conjugation by an element of $F_{4}(K)$, hence the
subgroups $(2^{3})_{a}$, $(2^{4})_{b}$, and $(2^{5})_{b}$ with distributions
$2{\rm B}_{7}$, $2{\rm B}_{14}{\rm C}_{1}$ and $2{\rm B}_{28}{\rm C}_{3}$ have
the same normaliser quotient as in $E_{6}(K)$, and in $F_{4}(K)$. For
$E=(2^{3})_{b}$, any automorphism induced by conjugation in $G$ must preserve
the subgroup $\left<x_{1},x_{2}\right>$ as this is generated by all $2{\rm
B}$-involutions in $E$. The group $\operatorname{SL}_{3}(2)$ of automorphisms
of $(2^{3})_{a}$ centralises $z$, and the stabiliser of
$\left<x_{1},x_{2}\right>$ in $\operatorname{SL}_{3}(2)$ induces all such
automorphisms on $(2^{3})_{b}$. It follows that
$N_{G}(E)/C_{G}(E)=2^{2}.\operatorname{SL}_{2}(2)$ is the stabiliser of a
$2$-dimensional subspace.

A direct calculation shows that if $E=(2^{4})_{a}$ or $E=(2^{4})_{c}$, then
$E$ contains a unique $2{\rm B}$-pure subgroup of order $8$, namely
$(2^{3})_{a}$, and so $N_{G}(E)\leqslant
N_{G}((2^{3})_{a})=((2^{3}\mathpunct{:}\operatorname{SL}_{3}(2))\times{\rm
Sp}_{6}(K)$. Since $y_{1}$, $z$, and $y_{1}z$ are involutions in ${\rm
Sp}_{6}(K)$ lying in distinct $G$-conjugacy classes, and since ${\rm
Sp}_{6}(K)$ is centralised by the action of $\operatorname{SL}_{3}(2)$, it
follows that
$N_{G}((2^{4})_{a})/C_{G}((2^{4})_{a})=N_{G}((2^{4})_{c})/C_{G}((2^{4})_{c})=\operatorname{SL}_{3}(2)$.
The group $E=(2^{4})_{d}$ contains a unique subgroup $E_{0}$ with distribution
$2{\rm B}_{6}{\rm C}_{1}$, and this in turn contains a unique involution in
class $2{\rm C}$; thinking of $N_{G}(E)/C_{G}(E)$ as a subgroup of
$\operatorname{GL}_{4}(2)$, it is contained in the set of matrices of the form

$\begin{pmatrix}1&\mathbf{0}&\mathbf{0}\\\ \ast&A&\mathbf{0}\\\
\ast&\ast&1\end{pmatrix}$

with $A\in\operatorname{GL}_{2}(2)$. The $N_{G}(M)$-stabiliser of
$(2^{4})_{d}$ induces all such automorphisms, giving the stated normaliser
structure $N_{G}(E)/C_{G}(E)=2^{1+4}.\operatorname{Sym}_{3}$. Similarly,
$E=(2^{5})_{a}$ contains a unique subgroup $E_{0}$ with distribution $2{\rm
B}_{14}{\rm C}_{1}$ and $N_{G}(E)$ preserves the direct-sum decomposition
$E=E_{0}\times\langle z\rangle$. Moreover, $E_{0}$ contains a unique
involution in class $2{\rm C}$. Hence $N_{G}(E)/C_{G}(E)$ can be identified
with a set of matrices of the form

$\begin{pmatrix}1&\mathbf{0}&\mathbf{0}\\\ \ast&A&\mathbf{0}\\\
\mathbf{0}&\mathbf{0}&1\end{pmatrix}$

with $A\in\operatorname{GL}_{3}(2)$. Again, $N_{G}(M)$ induces all such
automorphisms, hence $N_{G}(E)/C_{G}(E)=2^{3}:\operatorname{SL}_{3}(2)$. If
$E=(2^{5})_{c}$, then $E$ contains a unique subgroup $E_{0}$ with distribution
$2{\rm B}_{14}{\rm C}_{1}$, and $E_{0}$ is a conjugate of $(2^{4})_{b}$. Thus,
$N_{G}(E)\leqslant N_{G}(E_{0})=(2^{3}\times\operatorname{SL}_{2}(K)\times{\rm
Sp}_{4}(K)).(2^{3}\mathpunct{:}\operatorname{SL}_{3}(2))$. The subgroup
$\operatorname{SL}_{2}(K)$ centralises $M$ and thus $E$, while the subgroup
${\rm Sp}_{4}(K)$ contains an element swapping $y_{1}$ and $y_{2}$ (taking
$y_{1}y_{2}$ to be the central element of ${\rm Sp}_{4}(K)$ without loss of
generality), and acting trivially on the subgroup
$\left<x_{1},x_{2},x_{3}\right>$ of $E$. Hence $N_{G}(E)/C_{G}(E)\leqslant
2\times(2^{3}\mathpunct{:}\operatorname{SL}_{3}(2))$. We again find that
$N_{G}(M)$ induces all such automorphisms on $E$, so
$N_{G}(E)/C_{G}(E)=2\times(2^{3}:\operatorname{SL}_{3}(2))$. Finally, let
$E=(2^{5})_{d}$. This contains a unique subgroup $E_{0}$ with distribution
$2{\rm B}_{12}{\rm C}_{3}$, and $E_{0}$ in turn contains a unique subgroup
with distribution $2{\rm C}_{3}$. Thus $N_{G}(E)/C_{G}(E)$ is contained in the
set of all matrices of the form

$\begin{pmatrix}1&\mathbf{0}&\mathbf{0}\\\ \ast&A&\mathbf{0}\\\
\ast&\ast&B\end{pmatrix}$

with $A,B\in\operatorname{GL}_{2}(2)$. Once again, $N_{G}(M)$ induces all such
automorphisms on $E$, so $N_{G}(E)/C_{G}(E)$ is determined.

###  6.7. Proof of Proposition 6.3 for $G=E_{7,{{\text{\rm ad}}}}(K)$

Let $G=E_{7,{{\text{\rm ad}}}}(K)$ and $p=2$. The three classes of involutions
in $G$ are denoted by $2{\rm BC}$, $4{\rm A}$, and $4{\rm H}$, corresponding
to the classes in the simply connected cover of $G$ which map onto them (see
Section 3.3). In [40], Yu has identified 78 classes of elementary abelian
$2$-subgroups of $G$ and determined their normaliser quotients. However, it
remains to determine their centraliser structure and whether or not these
subgroups are toral. In the course of determining this information, we have
found some typographic errors in the ancillary data of [40], particularly in
[40, Remark 7.27], although we agree with the eventual classification itself.
For this reason, we now carry out a number of our own calculations, verifying
much of the information in [40].

The toral elementary abelian $2$-subgroups of $G$ can be determined with our
algorithm in Section 5.1. We find that there are $32$ conjugacy classes of
such toral subgroups. It follows from [22] that $G$ has two maximal non-toral
subgroups up to conjugacy, of orders $2^{7}$ and $2^{8}$, respectively.
Between them, these contain representatives of each class of non-toral
subgroups; since the subgroup of order $2^{8}$ is obtained by adjoining a
Chevalley involution to a maximal toral subgroup of order $2^{7}$, it also
contains a representative of every class of toral subgroups.

By [22, Table 1], the maximal non-toral subgroups $2^{7}$ and $2^{8}$ are each
self-centralising, and have normaliser quotients as follows

$N_{G}(2^{8})/2^{8}=2^{7}:\operatorname{SO}_{7}(2)\quad\text{and}\quad
N_{G}(2^{7})/2^{7}=(2^{2\cdot 2}\times 2^{2\cdot
3})\mathpunct{:}(\operatorname{Sym}_{3}\times\operatorname{Sym}_{3}\times\operatorname{SL}_{3}(2)).$

Because $N_{G}(2^{7})$ and $N_{G}(2^{8})$ are finite, and their actions on
$2^{7}$ and $2^{8}$ are known (cf. [40, Proposition 7.27] and Appendix A.2),
we can explicitly construct these groups as matrix groups of degree $7$ and
$8$, respectively. In particular, we can calculate the orbits of these groups
on the subgroups of $2^{7}$ and $2^{8}$, using Magma [13]. Our computation
also returns the complete subgroup lattice for these groups. Appendix A.3
comments on our construction. Recall that every non-toral subgroup of $G$
appears as a subgroup of at least one of the maximal $2^{7}$ and $2^{8}$. The
(known) distributions of $2^{7}$ and $2^{8}$ allow us to identify the
distributions of the subgroups in each orbit. From this, together with the
information on toral subgroups already calculated, we can determine for most
subgroups occurring whether the subgroup is toral or non-toral. A few cases
require further argument, and for these we defer to information from [40]. In
the end, we are able to identify $46$ conjugacy classes of non-toral
elementary abelian $2$-subgroups of $G$; these are listed in Table 6. We label
these groups according to the classification in [40], see Appendix A.1 for a
description of this classification. Our investigations yield the following
corollary.

######  Corollary 6.5.

The complete Hasse diagram of non-toral $2$-subgroups in $G$ is given in
Figure 1.

The structure of the normaliser quotients of the groups in Table 6 has been
determined in [40, Proposition 7.26], see also Appendix A.2 for more details.
In the remainder of this section, we discuss the centralisers of the non-toral
subgroups; for this the inclusions implied in the Hasse diagram (Figure 1)
will be useful. In most cases, the centraliser of $E$ with $|E|>2^{2}$ is
determined in $C_{G}(U)$ where $E=\langle U,u\rangle$ for some non-toral $U$;
the column labelled “$U$” in Table 6 lists this subgroup, see Case (2) in the
proof of Proposition 6.6.

######  Proposition 6.6.

If $E$ is as given in Table 6, then $C_{G}(E)$ is as given in that table.

###### Proof.

We consider the following sets of groups as defined in Table 6:

$\displaystyle\mathcal{X}_{0}$
$\displaystyle=\\{(2^{2})_{a},(2^{2})_{b},(2^{2})_{c},(2^{3})_{e},(2^{3})_{g}\\},$
$\displaystyle\mathcal{X}_{1}$
$\displaystyle=\\{(2^{4})_{j},(2^{5})_{k},(2^{5})_{f},(2^{5})_{h},(2^{6})_{g},(2^{6})_{c},(2^{6})_{b}\\},$
$\displaystyle\mathcal{X}_{2}$
$\displaystyle=\\{(2^{4})_{i},(2^{5})_{i},(2^{5})_{j},(2^{6})_{f}\\},$

and make a case distinction.

  * (Case 1)

Suppose $E\in\mathcal{X}_{0}$. The centraliser of each Klein four-subgroup of
a compact simple real Lie group of type $E_{7}$ is given in [24, Table 6], and
in our three cases $(2^{2})_{a},(2^{2})_{b},(2^{2})_{c}$, this is the direct
product of the non-toral subgroup in question with a maximal connected
subgroup (in the real topology) of the centraliser in the complex algebraic
group; cf. [21, Table 4.3.3]. (Note that [24, Table 6] lists the centraliser
$\operatorname{Aut}(\mathfrak{e}_{7})^{\Gamma}$ of each Klein-four subgroup
$\Gamma$ of $\operatorname{Aut}(\mathfrak{e}_{7})$, and the latter is a
maximal compact subgroup of $G$ since $G$ is adjoint.) The centraliser
structure for the subgroups of rank $2$ in Table 6 follows.

To consider $(2^{3})_{e}$ and $(2^{3})_{g}$, we need some preliminary results.
Let $H=E_{7,{\text{\rm sc}}}$, so that $G=H/Z$ where $Z=Z(H)=\langle
z\rangle$. By [21, Tables 4.3.1–4.3.3] there is an involution $t\in H$ such
that

$C_{H}(t)=\operatorname{Spin}_{12}(K)\circ_{2}\operatorname{SL}_{2}(K),$

where the central product is over $2=\langle
2_{s}{:}2\rangle\leqslant\operatorname{Spin}_{12}(K)\times\operatorname{SL}_{2}(K)$.
Note that $Z\leqslant Z(C_{H}(t))$, and $Z(C_{H}(t))=\langle
2_{c}{:}1,2_{s}{:}1\rangle=Z(\operatorname{Spin}_{12}(K))$ with
$2_{s}{:}1=1{:}2$ generates $Z(\operatorname{SL}_{2}(K))$. It follows also
from [21, Tables 4.3.1–4.3.3] that $tZ\in G$ has centraliser
$C_{G}(tZ)=\operatorname{Spin}_{12}(K)\circ_{2^{2}}\operatorname{SL}_{2}(K)$,
where the central product is over $2^{2}=\langle 2_{s}{:}1,2_{c}{:}2\rangle$.
This shows that $C_{G}(tZ)=C_{H}(t)/Z$. Since $Z(C_{G}(tZ))=2$ is generated by
$2_{c}{:}1$, it follows that $Z=\langle 2_{s}{:}1\rangle$ is contained in
$\operatorname{Spin}_{12}(K)\circ_{2}\operatorname{SL}_{2}(K)$, so
$z=2_{s}{:}1=1{:}2\in Z(\operatorname{SL}_{2}(K))$. A direct computation shows
that there is a toral $X=2^{2}\leqslant H$ with $t\in X$ such that
$C_{H}(X)^{\circ}=A_{5}T_{2}$ and
$C_{H}(X)=C_{H}(X)^{\circ}.w=C_{H}(X)^{\circ}.2$, and
$\operatorname{Out}_{H}(X)=\operatorname{Sym}_{3}$. Let $X=\langle t,y\rangle$
for some involution $y=y_{1}{:}y_{2}\in C_{H}(t)$. Since
$C_{C_{H}(t)}(y)^{\circ}=A_{5}T_{2}$, it follows that
$C_{\operatorname{SL}_{2}(K)}(y_{2})^{\circ}=T_{1}$ and
$C_{\operatorname{Spin}_{12}(K)}(y_{1})^{\circ}=A_{5}T_{1}$.

By [21, Tables 4.3.1–4.3.3], there is a projective involution
$u_{1}\in\operatorname{Spin}_{12}(K)$ such that
$C_{\operatorname{Spin}_{12}(K)}(u_{1})=\operatorname{SL}_{6}(K)\circ_{3^{*}}T_{1}$,
where $3^{*}=\gcd(3,\ell-1)$ with $\ell$ being the characteristic of $K$.
Choose a subgroup $\langle
u_{2},v_{2}\rangle\leqslant\operatorname{SL}_{2}(K)$ isomorphic to $Q_{8}$
with $u_{2}^{2}=v_{2}^{2}=z$. By [21, Tables 4.3.1–4.3.3], we may suppose
$y_{2}=u_{2}$, and so $y_{1}^{2}=y_{2}^{2}=z$; this shows that
$y_{1}\in\operatorname{Spin}_{12}(K)$ is a projective involution as well. By
[21, Tables 4.3.1-4.3.3], we may suppose $y_{1}=u_{1}$, so

$C_{H}(X)=(T_{2}\circ_{3^{*}}\operatorname{SL}_{6}(K)).w.$

Now write $w=w_{1}{:}w_{2}\in C_{H}(t)$ with
$w_{1}\in\operatorname{Spin}_{12}(K)$ and $w_{2}\in\operatorname{SL}_{2}(K)$.
It follows from [21, Table 4.3.1] that $w_{2}$ acts via inversion on $T_{1}$;
note that $w_{2}$ normalizes $C_{\operatorname{SL}_{2}(K)}(y_{2})=T_{1}$. This
implies that $w_{2}$ centralises $C_{\operatorname{SL}_{2}(K)/Z}(y_{2}Z)$, so
$w_{1}$ centralises $C_{\operatorname{Spin}_{12}(K)/Z}(y_{1}Z)$. By [21, Table
4.3.1], the element $w_{1}$ induces $\gamma{:}i$ on
$\operatorname{SL}_{6}(K)\circ_{3^{*}}T_{1}$; here $\gamma$ and $i$ denote a
graph automorphism and inversion, respectively. It follows that $w$ acts via
inversion on $T_{2}$ and induces a graph automorphism on
$\operatorname{SL}_{6}(K)$. Note that
$\operatorname{Out}_{H}(X)=\operatorname{Sym}_{3}$ acts on both
$Z(C_{H}(X)^{\circ})=2\times T_{2}$ and $Z(\operatorname{SL}_{6}(K))=2\times
3^{*}$. Moreover, we have $X\leqslant 2\times(T_{2})_{(2)}$, and it follows
that $\operatorname{Sym}_{3}$ centralizes $Z(\operatorname{SL}_{6}(K))_{2}=2$.
If $Z\neq Z(\operatorname{SL}_{6}(K))_{2}$, then $\operatorname{Sym}_{3}$
centralises at least two distinct involutions in $2\times(T_{2})_{(2)}$, which
is impossible; this proves that $Z\leqslant Z(\operatorname{SL}_{6}(K))$. Let
$Y=XZ=2^{3}$, so that $C_{H}(Y)=C_{H}(X)$. Note that $Y/Z$ is toral in $G$ and
$C_{G}(Y/Z)\geqslant C_{H}(X)/Z$. A direct computation shows that
$Y/Z=2^{2}=2{\rm B}{\rm C}_{3}$ and it yields that

$C_{G}(Y/Z)=(T_{2}\circ_{3^{*}}(\operatorname{SL}_{6}(K)/Z)).g,$

where $g=wZ$ acts via inversion on $T_{2}$ and induces a graph automorphism on
$\operatorname{SL}_{6}(K)$.

Now consider $E=\langle Y/Z,x\rangle=2^{3}$ for some $x\in C_{G}(Y/Z)\setminus
C_{G}(Y/Z)^{\circ}$; in particular, $C_{T_{2}}(x)=2^{2}$, and $x$ induces a
graph automorphism on $\operatorname{SL}_{6}(K)$. By [21, Tables 4.3.1–4.3.3],
there are two graph automorphisms, $\gamma_{1}$ and $\gamma_{2}$, such that
$C_{\operatorname{SL}_{6}(K)}(\gamma_{1})={\rm Sp}_{6}(K)$ and
$C_{\operatorname{SL}_{6}(K)}(\gamma_{2})=\operatorname{SO}_{6}(K)$, and
$C_{{\rm PSL}_{6}(K)}(\gamma_{1})={\rm PSp}_{6}(K)$ and $C_{{\rm
PSL}_{6}(K)}(\gamma_{2})={\rm PSO}_{6}(K).\gamma$. Note that each group
$\langle Y/Z,\gamma_{i}\rangle$ is non-toral. Comparing dimensions, we can
therefore suppose that

$(2^{3})_{g}=\langle
Y/Z,\gamma_{1}\rangle\quad\text{and}\quad(2^{3})_{e}=\langle
Y/Z,\gamma_{2}\rangle;$

in particular, we can assume that $Y/Z$ is a subgroup of $(2^{3})_{g}$ and
$(2^{3})_{e}$. If $3^{*}=1$, then it follows from above that
$C_{G}(Y/Z)=(T_{2}\times{\rm PSL}_{6}(K)).x$, so
$C_{G}((2^{3})_{g})=2^{3}\times{\rm PSp}_{6}(K)$ and
$C_{G}((2^{3})_{e})=2^{3}\times{\rm PSO}_{6}(K).\gamma$, as claimed. If
$3^{*}=3$, then $Z(\operatorname{SL}_{6}(K))=U=6$ and each
$C_{U}(\gamma_{i})=Z$. Write $V=U/Z$ and note that

$C_{\operatorname{SL}_{6}(K)}(\gamma_{1})/U\leqslant
C_{\operatorname{SL}_{6}(K)/Z}(\gamma_{1})/V\leqslant C_{{\rm
PSL}_{6}(K)}(\gamma_{1}).$

This yields $C_{\operatorname{SL}_{6}(K)}(\gamma_{1})/U={\rm
PSp}_{6}(K)=C_{{\rm PSL}_{6}(K)}(\gamma_{1})$, and it follows that
$C_{\operatorname{SL}_{6}(K)/Z}(\gamma_{1})={\rm Sp}_{6}(K)/Z={\rm
PSp}_{6}(K)$ and $C_{G}((2^{3})_{g})=2^{3}\times{\rm PSp}_{6}(K)$, as claimed.
Similarly, we deduce that

${\rm PSO}_{6}(K)\leqslant
C_{\operatorname{SL}_{6}(K)/Z}(\gamma_{2})\leqslant{\rm PSO}_{6}(K).\gamma.$

Let $h\in\operatorname{SL}_{6}(K)/Z$ such that $hV=\gamma\in C_{{\rm
PSL}_{6}(K)}(\gamma_{2})={\rm PSO}_{6}(K).\gamma$. Since $\gamma^{2}$ acts
trivially on ${\rm PSO}_{6}(K)$, it follows that $h^{2}\in
Z(\operatorname{SL}_{6}(K)/Z)=V$, and so $h$ has order $2$ or $6$. Replacing
$h$ by $h^{3}$, if necessary, we may suppose that $|h|=2$. Now
$(hV)^{\gamma_{2}}=hV$, and so $h^{\gamma_{2}}\in hV$. Since
$|h^{\gamma_{2}}|=2$, it follows that $h^{\gamma_{2}}=h$ and therefore
$C_{\operatorname{SL}_{6}(K)/Z}(\gamma_{2})={\rm PSO}_{6}(K).\gamma$. This
shows that $C_{G}((2^{3})_{e})=2^{3}\times{\rm PSO}_{6}(K).\gamma$, as
claimed.

  * (Case 2)

Suppose $E\not\in_{G}\mathcal{X}_{0}$. It follows from Figure 1 that
$E=\langle U,u\rangle$ for some non-toral elementary abelian subgroup $U$
containing some subgroup $V\in\mathcal{X}_{0}$, and an involution $u$ in
$C_{G}(U)\setminus U$. In particular,

$C_{G}(E)\leqslant C_{G}(U)\leqslant C_{G}(V)=2^{i}\times N.c$

for some $i\in\\{2,3\\}$ and $N\in\\{{\rm PSO}_{6}(K),{\rm
PSp}_{6}(K),F_{4}(K),{\rm PSp}_{8}(K),{\rm P}\Omega_{8}(K)\\}$ and
$c\in\\{1,\gamma\\}$, see Table 6. Thus, $C_{G}(U)^{\circ}$ is a reductive
group and $u$ acts on $C_{G}(U)^{\circ}$. The centralisers of involutions of
$C_{G}(U)^{\circ}$ can be determined by [21, Tables 4.3.1–4.3.3]; note that
$C_{G}(E)=C_{C_{G}(U)}(u)$ and $\dim C_{G}(E)=\dim C_{C_{G}(U)^{\circ}}(u)$.
We use this information to determine $C_{G}(E)$; the subgroups $U$ we use are
listed in Table 6.

  * (Case 2a)

First suppose $E\not\in_{G}\mathcal{X}_{1}\cup\mathcal{X}_{2}$. By the choice
of $U$, it follows from [21, Tables 4.3.1–4.3.3] that $u$ is uniquely
determined, up to conjugacy in $C_{G}(U)$, by the dimension $\dim
C_{C_{G}(U)^{\circ}}(u)$, and so $C_{G}(E)=C_{C_{G}(U)}(u)$ can be determined
by [21, Tables 4.3.1–4.3.3]. For example, if $E=(2^{3})_{f}$, then $C_{G}(E)$
has dimension $12$. By Figure 1, we have $E=\langle U,u\rangle$ with
$U=(2^{2})_{c}$ and $C_{G}(U)=2^{2}\times{\rm P}\Omega_{8}(K)$. It follows
from [21, Tables 4.3.1–4.3.3] that $\text{P}\Omega_{8}(K)$ contains a unique
involution $u$ such that $\dim C_{P\Omega_{8}(K)}(u)=12$. In fact, $C_{{\rm
P}\Omega_{8}(K)}(u)=(\operatorname{SO}_{4}(K)\circ_{2}\operatorname{SO}_{4}(K))).\langle\gamma{:}\gamma,\leftrightarrow\rangle$,
thus

$C_{G}(E)=2^{2}\times(\operatorname{SO}_{4}(K)\circ_{2}\operatorname{SO}_{4}(K))).\langle\gamma{:}\gamma,\leftrightarrow\rangle=2^{2}\times(\operatorname{SO}_{4}(K)\circ_{2}\operatorname{SO}_{4}(K))).2^{2}.$

The centraliser structure of the other groups
$E\not\in_{G}\mathcal{X}_{1}\cup\mathcal{X}_{2}$ can be obtained similarly; we
give a few details for $(2^{4})_{x}$ with $x\in\\{a,c,e,h\\}$:

    * $\bullet$

We have $(2^{4})_{c}=\langle(2^{3})_{b},u\rangle$ with
$u\in\operatorname{Spin}_{9}(K)$; since $\operatorname{Spin}_{9}(K)$ is simply
connected, the centraliser
$C_{\operatorname{Spin}_{9}(K)}(u)=\operatorname{Spin}_{8}(K)$ is connected,
hence $C_{G}((2^{4})_{c})=2^{2}\times\operatorname{Spin}_{8}(K)$.

    * $\bullet$

We have $(2^{4})_{e}=\langle(2^{3})_{d},u\rangle$ and
$C_{G}((2^{3})_{d})=2^{2}\times({\rm Sp}_{4}(K)\circ_{2}{\rm Sp}_{4}(K)).2$.
Consider the element $v=I_{2}\times(-I_{2})\in{\rm Sp}_{4}(K)$ and let $w$ be
a permutation matrix corresponding to $(1,3)(2,4)$, so that $[v,w]=-I_{4}\in
Z({\rm Sp}_{4}(K))$, $C_{{\rm Sp}_{4}(K)}(v)=(\operatorname{SL}_{2}(K))^{2}$,
and $w$ swaps the two factors of $C_{{\rm Sp}_{4}(K)}(v)$. If $u=vv\in{\rm
Sp}_{4}(K)\circ_{2}{\rm Sp}_{4}(K)$, then the outside 2 of $({\rm
Sp}_{4}(K)\circ_{2}{\rm Sp}_{4}(K)).2$ centralises $u$, and so does and
$w{:}w\in{\rm Sp}_{4}(K)\circ_{2}{\rm Sp}_{4}(K)$; hence
$C_{G}((2^{4})_{e})=2^{2}\times((\operatorname{SL}_{2}(K))^{2}\circ_{2}(\operatorname{SL}_{2}(K))^{2}).2^{2}$.

    * $\bullet$

We have $(2^{4})_{h}=\langle(2^{3})_{c},u\rangle$ with
$C_{G}((2^{3})_{c})=2^{2}\times(\operatorname{GL}_{4}(K)/Z).\gamma$. Let
$v=\operatorname{diag}(s,s,s^{-1},s^{-1})$ be in $\operatorname{GL}_{4}(K)$
with $|s|=4$ and let $w\in\operatorname{GL}_{4}(K)$ be the permutation matrix
corresponding to $(1,3)(2,4)$. Then
$C_{\operatorname{GL}_{4}(K)}(v)=\operatorname{GL}_{2}(K)^{2}$ and
$[v,w]=-I_{4}\in Z(\operatorname{GL}_{4}(K))$. Note that the graph
automorphism $\gamma$ in $(\operatorname{GL}_{4}(K)/Z).\gamma$ also gives
$[v,\gamma]=-I_{4}$; if $u=vZ$ and $y=wZ$, then
$C_{G}((2^{4})_{h})=2^{2}\times((\operatorname{GL}_{2}(K))^{2}/Z).y.\gamma$.

    * $\bullet$

We have $(2^{4})_{a}=\langle(2^{3})_{a},u\rangle$ with
$u\in\operatorname{SL}_{2}(K)\circ_{2}{\rm Sp}_{6}(K)$. Write