i4.7. FINISHING Ls(2) WITH (vG1) ABELIAN 1059(b) Lt centralizes Kt.
(c) Pt projects faithfully on both Pj( and PJ.
In case (a), Pj( = Pt ::; Lt ::; Kt, and hence PJ centralizes Lt, so that
p+::; LtPJ::; Nr+(Lt), contrary to an earlier observation. In case (b), PJ =
Pt ::; Lt, and Pj( centralizes Lt, so p+ ::; P'J<Lt ::; Nr+(Lt), for the same
contradiction. Therefore case (c) holds. We saw that either T normalizes Kt or
(KtT/ ~ 83 x 83. However the latter case is impossible, as then by A.1.31.1,
p+ ::; 031 (J+) = 0( (KtT/ ), contradicting Lt not subnormal in J+. Thus T
acts on Kr, so as Tacts on Li, it acts on the projections L"j{ and Lt of Lt on
Kt and Cr+ (Kt), respectively. Then as T acts on Li = PL0 2 (Li), and Pt ::;
L"j{L6 = 02(L"j{)02(Lt)P+, p+ normalizes 02(L"j{)02(Lt)Pt = Lt, for the
same contradiction yet again. This finally completes the proof of 14.7.32. DLEMMA 14.7.36. If KE C(H), then K/02,z(K) is not sporadic.
PROOF. Assume K/02,z(K) is sporadic. By 14.7.32, KTLi E Hz, so without
loss H = KTLi. We conclude from 14.7.30 and F.9.18.4 that K* ~ M 22 or M 22.
As M22 and M22 have no FF-modules by B.4.2, 1 := [UH, K] is irreducible under K
using F.9.18.7. As q(H*, UH) ::; 2 by 14.5.18.3, B.4.2 and B.4.5 say that 1 is eitherthe code module for M 22 or the 12-dimensional irreducible for M 22. In either case
V of rank 2 lies in 1.
We first eliminate the case K* ~ M22, as in the proof of 13.8.21: First Li ::;
0
31
(H) = K by A.3.18. Since Li is solvable and normal in J :=Kn M, Jj0 2 (K)is a maximal parabolic of N/02(K) ~ A5/E 2 4. Then Cv(02(Li(T n K))) ::;
Cv(02(NT)), with m(Cv(02(NT))) = 1 by H.16.2.1. This is a contradiction,
since LiT induces GL(V) on V of rank 2in1, so that 02 (LiT) centralizes V.
Thus we may assume that K ~ M22· By 14.7.28, Li = Z(K) ::;I H*, so
H =KT, and 1 = UH = [UH, Li] = [UH,K] by 14.7.5.5. As L]' is inverted
in T*, H* = K*T* ~ Aut(M22). By 14.7.5.3, Q* E 8yl2(K*). By H.12.1.9,
m(CuH (T*)) = 1, so V2 = CuH (T*), and then V = [-fi2, Li]. Now H n M = NH(V)
by 14.3.3.6, so using H.12.1.7,
(H n M)* = NH·(V) ~ 85/E32/Z3.However from the structure of Aut(M 22 ), there is an overgroup Hi of L 1 T in
H (arising from the maximal parabolic of A5/ Ern/Z3 which is not contained in
85/E32/Z3) with Hi/02(Hi) ~ 83 x 83 and Hi i NH·(V) = (HnM)*, contrary
to Theorem 14. 7.29. DLEMMA 14.7.37. {1) ·(fH > (VCH•CV2)).
(2) UH is not the natural module for 02 (H*) ~ Ln(2), with 3::; n S 5.
{3) UH is not the natural module for H* ~ 87.PROOF. Set Ho := 02 (CH(V2)); by Coprime Action, H(J = 02 (CH*(V2)).
Assume that (1) fails; then UH= (VH^0 ). By 14.7.4.2, Ho acts on L 2 , so [L 2 , Ho] ::;
CL 2 (V2) = 02(L2), and then L2 acts on 02 (Ho02(L2))) =Ho. So as L2 also acts
on V, it acts on on (VH^0 ) =UH. But then LT= (LiT,L2) acts.on UH, so as
M = !M(LT), H ::; Na(UH)) ::; M, contrary to H E Hz. This contradiction
establishes ( 1).
If (2) fails, then cH.(V2) is irreducible on UH/V2, contrary to (1); so (2) holds.