612 4. PUSHING UP IN QTKE-GROUPSof groups of type Ru in chapter J of Volume I, Ki := Ki/Q1 ~ 85, and from J.2.3,
CQ 1 (X 1 ) ~ Q 8. Let v E Cv(X 1 ) - Vi; it follows that v* is of order 2 in CMi (XD,
so Mi~ D 12. Hence P 1 := Q 1 nM1 is of order 210 with [02(M1),X1]:::;; P1 and
ICp 1 (X1)I = 4. Then V3:::;; q>([02(M1),X1]):::;; 01(Q1), and 01(Q1) is the group
denoted by "U" in (Ru2). Thus by J.2.2.3, CQ 1 (X1) :::;; CQ 1 (U) :::;; CQ 1 (Vii). Hence
as ICQ 1 (X1)I = 8 > ICp 1 (X1)I, CK(Vii) 1:. M, as claimed.
Thus to complete the proof of Theorem 4.2.13, we may assume that case (6) of
Theorem C.4.8 holds, and it remains to derive a contradiction. Then LH is a block
of type M 24 or L 5 (2), and K ~ J 4. In particular, K is a component of the maximal
2-local H, and so centralizes 02 (H) # l. As Out(J4) = 1, H = K x CH(K), with
02(H):::;; CH(K). Hence I= LHNTnK(LH) x Cr(K), and setting Re:= CR(K),
R = 02 (!) = V x Re. As Vis self-centralizing in K, Re= CR(K) = CR(LH)· By
4.1.4.5, 02(H) :::;; R, so 02(H) :::;; Re.
Recall we reduced in the first two paragraphs of the proof to the case where
I:::;; LT. Thus as I= LHNrnK(LH) x Cr(K), I= LH(NrnK(LH)) x Re. Let S :=
NrnK(LH) x Re and ran involution in Z(Re); thus SE Syb(I) and r E Z(S).
Next 02 (!) = LH :::;; K :::;; CH(r) as r E Re, and hence r centralizes 02 (I)S =I,
so without loss HE M(Ca(r)). Then in particular K is a component of Ca(r).
From the structure of LH in case (6) of Theorem C.4.8, there is X of order
3 in LH with Cv(X) # l. Let Kx := CK(X)^00 and Gx := Ca(X). Then
Kx is quasisimple with Z(Kx) ~ Z5 and Kx/Z(Kx) ~ M22· Thus Kx is also
a component of Cax(r), and hence by I.3.2, Kx :::;; Lx E C(Gx) with Lx :=
Lx/O(Lx) quasisimple. We claim Kx = Lx, so assume that Kx < Lx. Then as
Kx E C(Cax (r)), r is faithful on Lx, and in particular on the quasisimple quotient
Lx. Now case (1.a) of Theorem A (A.2.1) holds since Lx is quasisimple, so Lx is
quasithin. Then inspecting the list of groups in Theorem B (A.2.2), we find that
none possesses an involutory automorphism r whose centralizer has a component
Rx which is a covering of M 22. This contradiction establishes the claim that
Kx = Lx E C(Gx)..
Recall from Hypothesis 4.2.3 that R:::;; R+ = Cr(M+/0 2 (M+)) = 02 ((L,T)),
and set Ri := NR+ (R) and Ri := Rif R. Recall also from our application of 4.2.12.2
early in the proof that L 5J. M, V:::;; LH, and V = [R 2 (LT), L], so Vis T-invariant.
If LH is ;:i,n L 5 (2)-block, then by Theorem C.4.8, V is one of the 10-dimensional
modules for LH /V, so as V is T-invariant, T induces inner automorphisms on
L/02(L). Of course T induces inner automorphisms on L/02(L) if LH is an M 24 -
block as Out(M24) = l. Thus LT= LR+, so as I< LT (since M = !M(LT)),
R = 02(I) < R+ and hence R < Ri. By 4.1.4.4, R = R+ n H, so as Ca(r) :::;; H we
have R = CR 1 (r). As R = V x Re, we can chooser E Re so that rV E CR;v(R 1 ).
Hence the map X : x* 1--+ [r, x] is an .LH-isomorphism of Ri with V: Since V :::;;
01(Z(R+)), the map is a homomorphism by a standard commutator formula 8.5.4
in [Asc86a]; then injectivity follows from R = CR 1 (r), and surjectivity as LH is
irreducible on V. Now there is v E Cv(X) - Cv(Kx), and for s E x-^1 (v) n Gx,
rs = rv. As M22 is not involved in the groups in A.3.8.2, Kx 5J. Gx, so as
[r, Kx] = 1, also [rs, Kx] = 1 and hence [v, Kx] = 1, contrary to the choice of v.
This contradiction completes the proof of Theorem 4.2.13. D