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
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 =
of a prime p, and when we write log q
we will mean logp q = a. Finally for such a q =
we denote by qn a primitive
prime divisor of
1, that is, a prime which divides
1 but not
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 =
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
for some l, where Ji denotes a Jordan block
of size i.
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