SATURATED FUSION SYSTEMS OF ESSENTIAL RANK 1

2.6 2-Dimensional linear groups
In this section F will always be a finite field of order q and V a 2-dimensional vector
space over F . Note that by 2.5.2,
Z(SL(V )) = Z(GL(V )) ∩ SL(V ) = 〈z〉,
where z is the linear transformation corresponding to multiplication with −1 ∈ F .
In particular, Z(SL(V )) = 1 if p = 2, and |Z(SL(V ))| = 2 if p is odd.
Lemma 2.6.1. Let p �= 2. Then there is exactly one involution in SL(V ), namely the
one in Z(SL(V )).
Proof. This is [15, 8.6.2]
Set V # = V \{0} and for v ∈ V # let
vF := {vλ : λ ∈ F }
be the F -subspace of V generated by v. We set for v ∈ V # , P (v) = CSL(V )(v). If
v, w is a basis of V , then we may view P (v) as the group of matrices
�� 1 0
λ 1
�
�
: λ ∈ F ,
with respect to v, w. In particular, P (v) ∼ = (F, +) and thus P (v) ∈ Sylp(SL(V )) =
Sylp(GL(V )). Moreover, CV (P (v)) = vF = CV (x) for every 1 �= x ∈ P (v), and
NGL(V )(P (v)) = NGL(V )(vF ). Observe that we may view NGL(V )(P (v)) as the group
of matrices
�� δ1 0
λ δ2
�
: δ1, δ2 ∈ F ∗ �
, λ ∈ F ,
with respect to a basis v, w. In the next lemma we describe the Sylow structure of
SL(V ) and GL(V ).
29