CHAPTER 2

Preliminaries

In this section we collect various results from the literature which will be needed

in our proofs.

Notation. First we introduce some notation for certain types of subgroups

in classical groups. Let G be a finite almost simple classical group with socle L

and associated vector space V . As usual, denote by Pi the parabolic subgroup

of G obtained by deleting the ith node of the standard Dynkin diagram; so Pi

is the stabilizer of a totally singular i-dimensional subspace of V , except when

L = P Ω2m(q)

+

and i = m − 1. In this last case there are two L-orbits on totally

singular m-spaces, Pm−1 and Pm being the stabilizers of representatives of the

different orbits. Also Pij denotes the intersection of two parabolic subgroups Pi

and Pj sharing a common Borel subgroup.

When L = Ln(q), denote by N1,n−1 the stabilizer of a pair of complementary

subspaces of V of dimensions 1,n − 1.

When L = Spn(q) with q even, write O for the normalizer in G of the natural

subgroup On(q) of L.

Now assume G is unitary, symplectic or orthogonal, and let W be a nonsingular

subspace of V of dimension i. We denote the stabilizer GW of W in G by

Ni,Ni+

or

Ni−

as follows:

GW = Ni, if G is unitary or symplectic, or if L = P Ω2m(q)

±

and i is odd;

GW = Ni ( = ±), if i is even, G is orthogonal and W has type O ;

GW = Ni ( = ±), if i is odd, L = P Ω2m+1(q) (q odd) and W

⊥

has type O .

For a classical subgroup H of G, we will sometimes write Pi(H),Ni(H), etc.

for the relevant parabolic subgroup Pi or nonsingular subspace stabiliser Ni in H.

Also q will always denote a power q =

pa

of a prime p, and when we write log q

we will mean logp q = a. Finally for such a q =

pa,

we denote by qn a primitive

prime divisor of

qn

− 1, that is, a prime which divides

pan

− 1 but not

pi

− 1 for

1 ≤ i an. By [47], such a prime exists except in the cases where (p, an) = (2, 6)

or an = 2, p + 1 =

2b.

The first two lemmas of this section concern the classification of involution

classes in symplectic and orthogonal groups in characteristic 2, and are taken from

[2, Sections 7,8].

Let V be a vector space of even dimension 2m over a finite field of character-

istic 2, and let ( , ) be a non-degenerate symplectic form on V with corresponding

symplectic group Sp(V ). For an involution t ∈ Sp(V ), define

V (t) = {v ∈ V : (v, t(v)) = 0}.

The Jordan form of t is (J2,J1 l

2m−2l)

for some l, where Ji denotes a Jordan block

of size i.

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