8.2. DETERMINING LOCAL SUBGROUPS, AND IDENTIFYING J 4 721If 02 (H+) is of order 3 or 5, then H =IT, so that (1) holds. Thus we may assume
02 (H+) ~ L 3 (2) or L2(31).
Let W be a chief section for LuH on 02 (LuiI) with [W, 02 (H)] =/:- 1 and set
(LuH)^1 := LuH/CLuH(W). As LuH is irreducible on W, 02 (H) = CH(W) and
02(Lu) ::; CLu (W). Then as 02 (H) centralizes Lu /0 2 (Lu ), H+ ~ H^1 centralizes
L~, and Wis the sum of isomorphic irreducibles for H^1 and for L~ by Clifford's
Theorem. Recall P = A EB B, with B either natural or a 5-dimensional indecom-
posable for Lu ~ S L2 ( 4). Thus we may choose W so th~t W is the sum of d ~ 2
copies of the natural module for L~, and Wis the tensor product of the naturalmodule for L~ with ad-dimensional 02 (H)-submodule D of W. As case (I) holds,
[ 02 (Lu H), VJ ::; V no;( H) = B, so [W, VJ is the image of B in W. Therefore Lu is
irreducible on [W, v+], so it follows that v+ induces a transvection on D. Therefore
Dis a natural module for 02 (H^1 ) ~ L 3 (2), which is impossible as H+ ~ Aut(L 3 (2))
and Wis a homogeneous £~-module. Therefore (1) is established.
Finally Vis T-invariant, and by (1) so is 02 (!) =AB, establishing (2). DLEMMA 8.2.10. (1) Lis a block of type L3(4)/9, M 2 2/10, or M24/ll.
(2) CT(L) = 1.
(3) V = 02(L).
(4) Z = Cv(T) is of order 2.
PROOF. By 8.2.9.3, V is not the code module for L ~ M 24. By 8.2.9.2, T
normalizes VA, so [02(L),AJ ::; 02 (£) n VA ::; VCA(V) = VU = V. ThenL = [L, A] centralizes 02(L)/V, so that (1) holds. By 3.2.10.9, Cz(L) = 1, so (2)
follows. By (1), [Z, L] ::; V. Then as the Sylow group T centralizes Z, we conclude
from (2) and Gaschiitz's Theorem A.1.39 that VZ = VGz(L) = V. Therefore
Z = Cv(T), so Z is of order 2, completing the proof of (4). By (1), L/V is
quasisimple, and as F*(L) = 02(L), Z(L/V) is a 2-group. Thus as the multiplier
of M 24 is trivial, (3) holds when L ~ M24; and similarly (3) holds when Z(L/V) = 1,
so we may assume that Z(L/V) =/:-1. If L ~ L 3 (4), we may consider a quotient of
L/V with center of order 2; then from the structure of the covering group in (3b)
of I.2.2, 02 (Lu)V/V is an indecomposable extension of a natural £ 2 (4) module
over a nonzero trivial submodule, which is not isomorphic to B as an Lu-module,
contrary to 8.2.8.5. Since an extension of M 22 over a center of order 2 restricts to
such an extension of £ 3 (4), this argument also eliminates extensions of M 22. This
completes the proof of (3). D
8.2.3. Constructing CG(z). At this stage, in view of 8.2.10.1, the cases re-
maining are
L3(4)/9, M22/l0, and M24/ll.
By 8.2.10.4, Z = Cv(T) is of order 2. In this section we let z denote a generator
of Z, and set C := Cc(z).
Using the subgroup of C generated by CM(z) and H (appearing essentially asKzT in the proof of 8.2.13), we will show that 02(G) is extraspecial with center
Z. Then using the fact that C is an SQTK-group, we eliminate the £ 3 (4)/9 and
M22/l0 cases, where C/02(C) is U4(2) or Sp5(2) in the shadows U5(2) or Co2.
This reduces us to the case where L/V ~ M 24 and Vis the cocode module. There
we show C has the structure of the centralizer of a 2-central involution in J 4 , which
allows us to identify G as J 4.