1172 i6. QUASITHIN GROUPS OF EVEN TYPE BUT NOT EVEN CHARACTERISTIC{8} L ~ J 2 and either r is inner with CL(r) ~ As/Qs * Ds or E4 x As, or r is
outer and CL(r) ~ Aut(L3(2)).
{9) L ~ J 4 , r is inner, and CL(r) ~ Aut(JYb)/D~ or Aut(M22)/E21i.{10) L ~HS, and eitherr is inner with CL(r) ~ Ss/(Q~*Z4) or Z2 xAut(A5),
orris outer with CL(r) ~ Ss or Ss/Ern.
{11) L ~Ru, r is inner, and CL(r) ~ Ss/E54/E32 or E4 x Sz(8).
PROOF. The Atlas [c+s5] contains a list of centralizers, as does [GLS98J.
Neither reference includes proofs for the sporadic groups, but there are proofs in
section 5 of chapter 4 of [GLS98] when L is of Lie type and odd characteristic.
Proofs for M24, He, and J2 appear in [Asc94], for Mn and Mi2 in [Asc03b], and
for HS in [Asc03a]. Proofs or references to proofs for the remaining groups can
be found in [AS76b]. D
LEMMA 16.1.6. Assume r is a 2-element of Aut(L) centralizing a Sylow 2-
subgroup of L. Then either
{1) r E Inn(L), and if L appears in case {i) of {E2) and r is an involution,
then either r is a long-root involution or L ~ Sp 4 (2n); or
{2) L ~ A 6 and r induces an automorphism in S 6 •PROOF. This is well known; it follows from 16.1.4 and 16.1.5 when r is of order
2. D
Our final preliminary results describe the possible embeddings among compo-
nents of involution centralizers.
LEMMA 16.1.7. Assume tis an involution in G, Lis a component of Ca(t), and
i is an involution in Cc( (t, L)). Set K := (LE(Ca(i))). Then one of the following
holds:
(1) K = L.
{2) L/02(L) ~ L2(2n), Sz(2n), or L2(P), p prime, K = KiKf with Ki a
component of Cc(i) and Ki I Ki, Kif0 2 (K 1 ) ~ L/0 2 (L), and L = CK(t)^00 •.{3) K is a component of Cc(i), K = [K, t], L is a component of CK(t), and
one of the following holds:
{a) K/02(K) ~ X(2^2 n), where X is a Lie type of Lie rank at most 2, but
not Sz(2n), U 3 (2n), or^2 F4(2n), and t induces a field automorphism on K/02(K)
with L/02(L) ~ X(2n)'.
{b} K ~ L 3 (2^2 n) for n > 1, t induces a graph-field automorphism on K,
and L ~ U 3 (2n).
(c) K/02(K) ~ L3(2n) for n > 1, t induces a graph automorphism on
K/02(K), and L ~ L2(2n).
{d) K ~ Sp4(2n), n > 1 odd, t induces a graph-field automorphism on K,
and L ~ Sz(2n).
L~A5.
{e) K ~ L4(2) or Ls(2), t induces a graph automorphism on K, and
(f) K/02(K) ~ M12 or J2 and L ~As.
(g) K/02(K) ~ h and L ~ L3(2).
{h) K/02(K) ~HS and L ~ A 6 or As.
{i) K/02(K) ~Ru and L ~ Sz(8).