722 8. ELIMINATING SHADOWS AND CHARACTERIZING THE J4 EXAMPLE
Let Lz := CL(z)^00 and C := C/Z.
LEMMA 8.2.11. (1) Lz E C(CM(z)).
(2) There exists an LzT-series 1 < Z < Vz < V with Vz := [V, 02(Lz)J, and Vzis the natural module for Lz/02(Lz) ~ L2(4), As, As.
(3) V/Vz is the A 5 -module, the core of the 6-dimensional permutation module,
the 4-dimensional irreducible, respectively.(4) 02 (Lz)/V induces the group of transvections on Vz with center Z, so
02(L/f') = 02(Lz) is LzT-isomorphic to the dual of Vz.
PROOF. Parts (2)-(4) followfromH.4.6.4, H.16.4, andH.15.3. Then (2) implies(1). D
Set E := (~^0 ).
LEMMA 8.2.12. (1) E ~ D8 is extraspecial, fore:= 4, 4, 6.
(2) E = 02(C).
(3) 02(Lz) =EV and Vz =En V.
PROOF. By 1.1.4.6, F(C) = 02 (C) =: Qc, so F(C) = Qc by A.1.8. There-
fore as Vz :SJ T, 1 -=J. Cv-JT):::; Z(Qc). Then as Lz is irreducible on Vz by 8.2.11.2,
Vz:::; Z(Qc), so E = (Vz^0 ):::; Z(Qc).
Let QM := 02 (LT). By parts (2) and (5) of 8.2.8, [V,Lu] = B. Then by
H.4.6.5, H.16.4.4, and H.15.8, Vz :::; B but Vz i U; therefore Vz"' :::; A but Vzh i U.
Thus as U =An QM and ~h:::; E, E i QM. But by 8.2.11.4, Lz is irreducible on
02(Lz'f') = 02(Lz), so E = 02(Lz). Thus as V = 02(£) by 8.2.10.3, EV= 02(Lz),
establishing the first statement in (3).
Now Z:::; V = 02 (£) with L irreducible on V, so if (QM) -=J. 1 then (QM) ~
V. But CLT(QM):::; QM, so each x of odd order in Lis faithful on QM/(QM),
whereas [QM,x]:::; V by 8.2.10.l. Thus (QM) = 1. Similarly as Z:::; V, Lis inde-
composable on QM. But by earlier remarks, Q ;;;nQ c :::; C QM ( E) :::; C QM ( 02 ( L z)).
Next from the structure of indecomposable extensions of V by a trivial quotient (ob-
tained from the duals of modules described in I.1.6), CQM ( 02(Lz)) :::; Cv-(02(Lz)),
while Cv-(02(Lz)) = Vz by H.4.6.6, H.16.4.5, and H.15.3.4. Hence Vz =QM n Qc.
Thus Vz = V n E, completing the proof of (3). Now using (3) we have
I
- 1+2e
IEI = IVzllE: Vzl = IVzllEV: V = IVzl · I02(Lz)I = 2
where e := 4, 4, 6. By 8.2.11.4, Z = CvJE), so (1) holds. (As e + 1 = m(Vz),
E ~ D8).
As E:::; Qc:::; 02(CLT(z)) = EQM, and Qc n QM= Vz:::; E, (2) holds. D
PROPOSITION 8.2.13. (1) V is the cocode module for L/V ~ M 24.
(2) L =Mand C/E ~ M22.2PROOF·. By 8.2.11.1, Lz E C(CM(z)), and of course Lz is T-invariant. Then
by 1.2.4, Lz :::; Kz E C(O), and the possibilities for Kz/02(Kz) are described in
A.3.12.
By 8.2.12.2, E = 02(C). Let C* := C/E. As Kz :Si C and E = 02 (C),
02 (K;) = 1; in particular Lz < Kz by 8.2.12.3. Indeed using that result, 02 (£;) ~
V /Vz is described in 8.2.11.3. We inspect the lists in A.3.12 and A.3.14 for such
subgroups, and conclude that Lis M24 and K; ~ M22; in particular, notice when