1196 16. QUASITHIN GROUPS OF EVEN TYPE BUT NOT EVEN CHARACTERISTIC
U = !1 1 (R) is of order 2, so as R ~Tc, m2(R) = 1 = m2(Tc). Recall u denotes
the involution in U and z the involution in Tc. The projection v of u on L is
2-central in LT, so conjugating in L if necessary, without loss v E Z(T), and then·
u E (z, v) =: E ~ Z(T). Now we may choose g as in 16.4.2.4, so that u = z^9 by
16.4.11.3, and g E Nc(T) by 16.5.2.1. Now
[T, T] = YL x Ye, (*)
where Ye is the preimage in Tc of [T/TL,T/TL], and YL is of index at most 2 in
the cyclic subgroup Y of order 4 in TL.
Assume that Ye= l; that is, that T/TL is abelian. Set Z := !11(Z(T)). Then
either Z = E, or Z = E(t) where t induces a transposition on L. Further Na(T)
centralizes [T, T] n Z = (v) by(*). However g does not centralize Z since u = z9,
so g ¢. T, and hence we may assume g has odd order. As g E Nc(T), g centralizes
v, so Z = E(t) = (v) x [Z, Nc(T)] where [Z, Nc(T)] is of rank 2, and g induces
an element of order 3 on Z. But then either z or u = zv lies in [Z, Nc(T)], so
as u = z9, we conclude z E [Z,Nc(T)], and then zz9 = z(zv) = v E [Z,Nc(T)],
whereas we saw Nc(T) centralizes v.
Therefore Ye -=f.1. Then as m2(Tc) = 1, (z) = !11(Yc), so by(*), !11([T,T]) =
E ~ !11(Z(T)). Therefore as g E Nc(T), g induces an element of order 3 on E,
and Nc(T) is transitive on E#. Thus as YL is cyclic, so is Ye by the Krull-Schmidt
Theorem A.1.15, and then Ye~ YL, so that Ye is cyclic of order at most 4.
Next Ye :::l T, so Y6 :::l T. Nows E TL - Y inverts Y and centralizes Ye, so
if IYcl = 4, thens does not act on Y6, a contradiction.
Thus Ye = (z), so YL = (v) as YL ~ Ye. As YL = (v), LT is A 6 or
- Assume L ~ A5. Then T =TL x Tc, so Tc ~ Qs since m 3 (Tc) = 1 and
(z) =Ye~ [T/TL,T/TL] ~[Tc, Tc]. But R =T!J, so Qs ~Tc~ R ~ R ~ T* =
TI,, impossible as TI, ~ D 8. Therefore LT ~ 86 , so T* ~ Z 2 x D 8. Then as
Tc~ R ~ R ~ T, while m2(Tc) = 1 and [T, Tc] -=f. 1, we conclude that Tc~ Z4
and t E T -Tc TL inverts Tc. As T* ~ Z2 x Ds, we may pick t so that t centralizes
TL and t^2 E Tc. Let Ti := Tc(t); then T = T 1 x TL, with T 1 ~ D 8 or Q 8 , and
TL ~ Ds. Now g E Nc(T) is transitive on E#; but this is impossible, as by the
Krull-Schmidt Theorem A.1.15, Nc(T) permutes {1>(T 1 Z(T)), 1>(TLZ(T))}. D
LEMMA 16.5.4. L* is not L 3 (4).PROOF. Assume otherwise. As U ~TI, and all involutions in Lare 2-central
in L from I.2.2.3b, u centralizes a Sylow 2-group of L. Then as R centralizes Tc,
we may take u E Z(TLTc).
Suppose first that U f;. Z(TL). Then Y := r 1 ,u(L) contains a maximal
parabolic P of L, and Y ~ H' by 16.4.2.5. If P ~ L', then P ~ Cc(U), so
U ~ CL(P) = 1, a contradiction. Thus Pf;. L', so K' has an £ 2 (4)-section, and
hence m2,3(H') > 2, contradicting G quasithin.
Therefore U ~ Z(TL), so U ~ Z 2 or E4. Now J(T) = J(TLTc) = TLJ(Tc) =
TLUc, where Uc= !11(Tc) ~ U. As u E Z(TcTL), u E Z(J(T)). Recall Uc E uc
so there is g E G with U^9 = Uc. Thus u9 E Uc ~ Z(J(T)), and by Burn-
side's Fusion Lemma A.1.35, we may take g EM:= Nc(J(T)). Hence g acts on
Z(J(T)) =: V, where V =Uc x VL, with VL := [V,X] ~ E 4 for X of order 3 in
NL(TL)· Now we argue, just as in the proof of Proposition 16.5.1, that Xis regular
in Vf, and Uc is a TI-set under M, so again by Proposition 14.2 in [GLS96],