558 2. CLASSIFYING THE GROUPS WITH IM(T)I =^1PROOF. By Notation 2.5.4, HE r+ so that H = KoS with Ka component
of H, Ko = (KH), and K/0 2 (K) is not a Bender group. Thus as K ~ H by
hypothesis, H = KS and K = 02 (H). Further by 2.5.8.5, for each 4-subgroup F
of K, NK(F) ~ S 4 , and if F :::; SK then CAutH(K)(F) :::; Auts(K). It follows as
H =KS with SE Syb(H) that if F:::; SK then NH(F) = NK(F)Cs(F), and inparticular Ns(F) E Syb(NH(F)).
Next as (U,Hu) E U*(H) by hypothesis, U E Syb(Hu) by 2.5.10.2. Hence
Hu = 02 (Hu)U E 'He with 02 (Hu) :::; 02 (H) = K. Set E := (z:u). Now
ZK E Z(S) :::; Z(U) by 2.5.10.3, so by B.2.14, E :::; 02(Hu) and E is elementary
abelian. In particular, E :::; U as U E Syl 2 (Hu ). As Hu :::; Gt by 2.5.11.2 and
Hu i. M but Ca(z) :::; M = !M(T), we conclude m(E) > 1. Then as 02 (H) :::; K
and m 2 (K) = 2, E ~ E 4. Now Hu:::; NH(E), and we saw in the previous paragraph
that NH(E) = NK(E)Cs(E), with NK(E) ~ S4 and Ns(E) E Syl2(NH(E)).
Since Hu i. M, A 4 ~ 02 (NK(E)) = 02 (Hu) and E = 02(0^2 (Hu)), so that(2) holds. Further NH(E) E 'He and U _::::; Ns(E) so that Ns(E) E f3 by 2.3.2.1.
Therefore (Ns(E),NH(E)) E U(H) and Ns(E) EU, so as (U,Hu) E U*(H), we
conclude Ns(E) = U E Syb(Hu), and hence NH(E) = 02 (Hu)Ns(E) = Hu.
This completes the proof of (1). Further (3) follows from (1) since we saw that
Ns(E) E Syl2(NH(E)).
Now assume that z E QE ~ Hu with QE a 2-group. Then as z E QE,
Na(QE) E 'He by 1.1.4.3. So as U E U, and Hu :::; Na(QE) with Hu i. M,
(U,Na(QE)) E U(Na(QE)) and Na(QE) Er. Then since (U,Hu) E U*(H) byhypothesis, we conclude U E Syb(Na(QE)) using 2.3.2.1. This completes the proof
of (4), and hence of 2.5.12. D
LEMMA 2.5.13. (1) INr(S) : SI = 2, and t"' = tz for each x E Nr(S) -S.
(2) If (zK) char S, or more generally if [x, ZK] = 1, then z = ZK and t"' = tzK·(3) If tzK E t^0 , then z = ZK and t"' = tzK.
PROOF. By 2.5.11.1, Z(S) = (z, t) ~ E4 with (z) = Z(T). By 2.5.11.2, S E
Syl2(Gt) and hence S = Cr(t), so (1) follows. Then (1) implies (2). Further z ~ t^0
by 2.5.11.2, so (1) also implies (3). D
REMARK 2.5.14. There are extensions of £ 4 (3) ~ PDci(3) by a 2-group, with
involution centralizer Z 2 x£ 3 (3) or Z 2 xAut(L 3 (3)), which are of even characteristic,
and in which a Sylow 2-group is contained in a unique maximal subgroup. The
first extension is even quasithin. The next lem.ma eliminates the shadows of such
extensions.
LEMMA 2.5.15. K is not Mu or £ 3 (3).PROOF. Assume otherwise. Then case (3) of 2.5.2 holds, and K = Ko ~ H
by 1.2.1.3. As z induces inner automorphisms on K, Kz := 02 (CK(z)) ~ SL 2 (3)
0.5. The structure of SQTK-groups
By 2.5.11.2, H =KS:::; Gt, so by 2.5.6, K ~Gt. Then K = 031 (Gt) by A.3.18.
By 2.5.11.1, Zs:= Z(S) = (z, t) ~ E4· Then as K = 031 (Gt), Kz = 031 (Ca(Zs)),
so x acts on Z(Kz) = (zK)· Hence by 2.5.13.2, z = ZK EK and t"' = tz.Next as Aut(Kz) is induced in KzS, we may choose x E Cr(Kz). Further-
more as (z) = CK(Kz), Mu = Aut(Mu), and IAut(L3(3) : £3(3)1 = 2 with
CAut(K)(Kz) ~ Z4 if K ~ £3(3), either:
(i) S induces inner automorphisms on K, and Cs(Kz) =Sax (z), or