1176 16. QUASITHIN GROUPS OF EVEN TYPE BUT NOT EVEN CHARACTERISTIC
Then since Ho ::; H < G and TL 1:. T£, conclusion (3) of 3.4 in [Asc75] holds:
namely Ai n A 2 =I= 1, and Ai is dihedral or semidihedral. But as 1 =/= Ai n A2 ::;
L n Lu::; 02 (L), L/0 2 (L) ~ Sz(8) by 16.2.5, so that Ai is of 2-rank at least 3 and
hence not dihedral or semidihedral. This contradiction completes the proof of (1).
As TL E Syb(L) and H permutes {L,Lu}, (1) implies (2). D
LEMMA 16.2.11. 02 (L) = 1, so L is tightly embedded in G.PROOF. If Lis not tightly embedded in G, then 1 =/= 02(L) = Z(L) by 16.2.8.2,
so to prove both assertions we may assume Z(L) =/= 1, and it remains to derive a
contradiction. By 16.2.5, L/Z(L) ~ Sz(8).
Set ZL := fh(TL)· From I.2.2.4, involutions ofTLZ(L)/Z(L) lift to involutions
of TL, and these involutions are the nontrivial elements of ZL, so ZL is elementary
abelian. Further Out(L) is of odd order. From these remarks we deduce:
(*) If A is an elementary abelian 2-subgroup of Nr(L) then A::; ZLTc and A
centralizes Z L.
Further as Z(L) =/= 1, m(ZL) > m(ZL/Z(L)) = 3.
By 16.2.10.2, there is g E G - H such that L9 n NH ( L) contains an involution i,
and as TE Syl2(H) we may take i ET. Then by(*), i centralizes ZL, so as Cc(i) ::;
H9 by 16.2.8.3, ZL ::; H9; then conjugating in H9, we may take ZL ::; T9. Hence
by (*), X := NzL (£9) centralizes V := ZJ,. Further IZL : XI ::; IH : NH(L)I =
2 < IZL : Z(L)I, and hence X 1:. Z(L). In particular 1 =/= X, so V::; Cc(X) ::; H
by 16.2.8.3. As X ::; ZL but X 1:. Z(L), NH(X) ::; NH(L), so V ::; NH(L).
Then as m(V) > 3 = m 2 (Aut(L)), Cv(L) =/= 1, so L ::; Cc(Cv(L)) ::; H^9 by
16.2.8.3. Similarly Cv(Lu) =/= 1 so that Lu::; H9, and then Lo ::; H9, contrary to
16.2.9.2. D
LEMMA 16.2.12. Tf n T ={TL, T£}.
PROOF. Assume otherwise. Then there is g E G-H with S := Tf::; T but S
is not equal to TL or T£. Now as ITLI > 2 =IT: Nr(L)I, 1 =/= Ns(L)::; Ns(TL); so
as L is tightly embedded in G by 16.2.11, S centralizes TL (and similarly T£) by
I.7.6 with G, L, TL, Tin the roles of "H, K, Q, S ". Then R := TL(z)::; Cc(S)::;
H^9 using 16.2.8.3. As R centralizes S E Syb(L9), we conclude from 16.1.6 that
R induces inner automorphisms 'on L9. Then as IRI = 2181, 1 =I= CR(L9), so
L^9 ::; Cc(CR(L^9 )) ::; H by 16.2.8.3. Similarly Lu9 ::; H, so Lg ::; H, contrary to
16.2.9.2. D
We are now in a position to complete the proof of Theorem 16.2.4.
By 16.2.10.2, there is g E G - H such that L9 n Nr(L) contains an involution
i. If i centralizes a Sylow 2-subgroup of L, we may assume by conjugating in L
that i centralizes TL. Then by 16.2.8.3, TL ::; Cc ( i) ::; H9, and conjugating in
H^9 we may assume TL ::; T^9. But now by 16.2.12, TL E {Tf, Tit^9 }, contrary to
L tightly embedded in G by 16.2.11 since g (j. H. Thus i does not centralize any
Sylow 2-subgroup of L. But as 02 (L) = 1 by 16.2.11, 16.2.5 says that L has one
conjugacy class of involutions, so we conclude i induces an outer automorphism on
L. Therefore by 16.1.4 and 16.1.5 applied to the list in 16.2.5, either
(i) L ~ L 2 (2^2 n), and i induces a field automorphism on L, or
(ii) L(i) ~ PGL2(p).