1202 16. QUASITHIN GROUPS OF EVEN TYPE BUT NOT EVEN CHARACTERISTIC
is irreducible on V and M normalizes X containing Z, V:::; X:::; L. Thus Tc:::; L
so Z(L) # 1, and hence L* is G 2 (4) and Z(L) = Tc is of order 2 by 16.1.2.2.
Then X/V ~ L 2 (4)/E 2 s and the chief factors for X/V on 02(X)/V are natural
modules. However Y := 071 (M) centralizes X/0 2 (X) as Aut(As) = 85, so since
the group of units of Endx;v(0 2 (X)/V) is GL 2 (4), Y centralizes 02(X)/V. Then
as V = (0 2 (X)), Y centralizes 02 (X) by Coprime Action, a contradiction as Y
induces Z 7 on V. Therefore u* is not a long-root involution.
If L ~ L 3 (2n) then all involutions of L are long-root involutions, so the
lemma is established in this case. Further when £* is of type G 2 ,^2 F4, or^3 D4,
the 2-central involutions are the long-root involutions, so the lemma holds in these
cases too. Thus we may assume L* ~ Sp 4 (2n), so that Z(L) = 1 by 16.1.2.1. As
u is not a root involution, u is 2-central of type c2 in£* by 16.1.4.2, so we may
take u E Z(T); thus the projection v of u is in Z(T), sou E Z(TLTc) since R
centralizes Tc. Proceeding as in the proof of 16.5.4, U :::; Z(TJ,). As U ~ U and
U* contains no root elements, m(U) :::; n. Also as in 16.5.4, J(T) = TLJ(Tc) =
TLUc, where Uc = D 1 (Tc) ~ U is elementary abelian. Let M := Na(J(T)).
Recall that Uc E u^0 , so by Burnside's Fusion Lemma A.1.35, Uc E UM. Now
V := Z(J(T)) = UcVL is elementary abelian of order rq^2 , where VL := Z(TL) ~
Eq2, q := 2n, and r := IUcl· Further M acts on (J(T)) = VL and on V, so as
Uc n VL = 1, also UCJ n VL = 1 for each m E M. Let (3 be the set of involutions
in V - VL either contained in Uc, or projecting on a member of VL - (Z1 U Z2),
where Z 1 and Z 2 are the two root groups in VL. Then
lf31 = (q^2 -1-2(q - 1) + l)(r -1) = ((q-1)^2 + l)(r - 1).
Let 'Y be the set of involutions contained in a member of U!f. If y E G such that
zY EH and zY is a root involution of L, then KY E .6. 0 by 16.4.9.3, contrary
to 16.5.10.5. It follows that 'Y ~ (3. Also L n M contains a Cartan subgroup Y
of NL(TL) of order (q - 1)^2 , and Y acts regularly on U*Y and hence also on UY.
Therefore as K is tightly embedded in G, and NM(K) normalizes VnK =Uc, Uc
is a TI-subset of V under the action of M by I.7.2.3, so
bl~ ((q - 1)^2 + l)(r -1) = lf31,
and hence as 'Y ~ (3, we conclude that 'Y = (3 and U!f = 'Y is of order 1 + (q -1)^2.
This is impossible since 1 + (q -1)^2 is even, while T:::; NM(Uc) and Tis Sylow in
G. This contradiction completes the proof of 16.5.13. D
We are now in a position to establish our main result Theorem 16.5.14.
By 16.5.12 and 16.5.13.2, we have reduced the possibilities for Lin (E2) to the
case where L ~ G2(2n)',^2 F4(2n)', or^3 D4(2n). By 16.5.10.1, n > 1 if£~ G 2 (2n),
and by 16.5.13.1, u is a short-root involution in£. By 16.1.2, either Z(L) = 1, or
L is G2(4) and Z(L) is of order 2. However in the latter case, from I.2.2.5b, u lifts
to an v element of order 4, sou= cv with c E Tc of order 4. This is impossible, as
CH (X) ~ E4, so (R) = 1 by 16.5.6.2, and hence R ~ R ~ Tc is elementary
abelian. Thus Z(L) = 1.
Let V be the root group of the projection v of u on L---except when L is
(^3) D 4 (2n), where we set V := Z(X). Then (cf. 16.1.4 and [AS76a] for further
details) one of the following holds: