548 2. CLASSIFYING THE GROUPS WITH JM(T)J = 1LEMMA 2.4.33. K = E(G2) and F*(G2) = K02(G2).Now suppose that U 2 ::::; Ca(K). For g E L2, zg E U2 ::::; Ca(z2) = G2, so K is
a component of Ca 2 (zg) by 2.4.33. By I.3.2 and 2.4.33, K::::; 02',E(Ca(zg)) =KB.
We conclude K = KB, and hence K = E(Ca(u)) for each u E uf. Therefore
Ks= E(Ca(u)) for each u E uf = u#. Also x centralizes Z1 and hence normalizes
Ks = E(Ca(z 1 )), so Ks= E(Ca(ux)) for each ux E (Ux)#. Further L = [L, U 0 ]by 2.4.30.4, so as U!J ::::; Ca(K), L = [L, ux]. Thus using the structure of K in
2.4.32,
K = (CK(u), CK(ux): u EU#) S:: Na(Ks).
As z 2 centralizes K, z 1 centralizes Ks, so K = [K,z1]::::; Ca(Ks), and hence
T = 8 1 (s) normalizes KKs = K x Ks. Let I:= KKsT. Since I contains L f:_
M = !M(T), 02 (J) = 1. As G is quasithin, m2,s(KK^8 ) ::::; 2, so K ~ Ls(2) rather
than A5. As 02(I) = 1, m 2 (T) ::::; m2(Aut(KKs)) = 4, so Q = Uo and R ~ Ds xDs.
It follows that RE 8yl 2 (KKs) and T = R(x, s), with x an involution inducing an
outer automorphism on K and Ks, and s an involution centralizing x. Then I has
5 classes of involutions, with representatives z, z2, x, s, and sx. Now 02(G2) ::::;Cs 1 (K) ~ D 8 , so 02 (G 2 ) centralizes 02 (G 2 )/(z 2 ) and z 2 , and hence by Coprime
Action also centralizes 02(G2). Therefore as F*(Ca 2 (K)) = 02(G2) using 2.4.33,we conclude that Ca 2 (K) is a 2-group, and hence Ca 2 (K) = Cs 1 (K) = 02(G2).
Thus G2/ K02( G2) ~ Out(K) which is a 2-group, so G 2 / K is a 2-group, and hence
K = 02 (G2), so ms(G2) = 1.
Now C1(s) ~ (s) x Ks(x) with Ks~ Ls(2), and the involutions in the subgroup
Ks diagonally embedded in K x Kx are in zG as z = z 1 z2; thus s ~ zf, since the
involutions in K = G2 are in zf Similarly sx ~ zf Next C1(x) = (x, s)(Ii x Jz)
·with Ii := CK(x) ~ 83 and If = fz. In particular as ms(G2) = 1, x ~ zf As
O(Ca 2 (x)) =/=-1, F*(Ca(x)) =/=-02(Ca(x)) by 1.1.3.2, so x ~ z^0.
But as G = 02 (G), by Thompson Transfer, x^0 n 8 =!=- 0. Therefore as we
saw x is not conjugate to z or z 2 , it must be conjugate to s. Arguing similarly
with 8 replaced by ( sx) uux, we conclude sx E s^0. So x^0 =. s^0 = ( sx) G, andhence by the previous two paragraphs, s, z, and z 2 are representatives for the
conjugacy classes of involutions of G. Thus s is in fact extremal in T: that is,
Ts:= CT(s) E 8yl2(Ca(s)). But each involution in C1(s) is fused in I to s, x, sx,
or z, so zf n Ts= 0. This is impossible as z2 E Ca(x) with x conjugate to s. This
contradiction shows U 2 f:_ Ca(K), and hence:
LEMMA 2.4.34. K = [K, U2].
Now U2 ::::; Ca 2 (L). But if K ~ Ls(2), then Ca 2 (L) = Ca 2 (K) from the
structure of Aut(K), so U2 centralizes K, contrary to 2.4.34. Therefore part (1) of
the following lemma holds:LEMMA 2.4.35. {1) K ~ A5, and some u E U 2 - (z 2 ) induces a transposition
on K centralizing L.
{2) The automorphism induced by x on K is not in 86.
For if part (2) of 2.4.35 fails, then setting (K81)+ := K81/CKs 1 (K), x+ EK+ R+,
so U~+ E utK. Then as Ut = 02(L+ R+) is weakly closed in R+ with respect to
K+ from the structure of A5, Uft = u~+, contrary to 2.4.30.4.LEMMA 2.4.36. {1) R = RK x RK with RK :=Rn KE 8ylz(K) ~ Ds.