1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

i4.7. FINISHING La(2) WITH (vGi) ABELIAN 1047 14.7.2. Eliminating solvable members of Hz· As was the case in Theorem 14.6.18 whe ...

1048 i4. L 3 (2) IN THE FSU, AND L2(2) WHEN .Cf(G, T) IS EMPTY REMARK 14. 7 .17. Notice ( 6) shows that our hypotheses are symme ...

14.7. FINISHING Ls(2) WITH (vG1) ABELIAN 1049 PROOF. Recall E = [U, Q] by definition, while [E, Q] = V from the action of Q* on ...

i050 i4. L 3 (2) IN THE FSU, AND L 2 (2) WHEN .Cf(G, T) IS EMPTY where Bo := C 8 M(L/02(L)). As Bi is Sylow in Li and Bi ::; Be ...

i4.7. FINISHING La(2) WITH (vG1) ABELIAN 105i unique noncentral Li -chief factor on W contained in R/V. So as R ::; CH(U) by 14. ...

1052 i4. L 3 (2) IN THE FSU, AND L2(2) WHEN .Cf(G, T) IS EMPTY c_ is the preimage in p_ of Cp_ (Li). As case (ii) of 14.7.20.2 h ...

14.7. FINISHING L 3 (2) WITH (VG1) ABELIAN 1053 Let Y E Syh(K), set X := Y n Li, and let I consist of the ¥-invariant subgroups ...

1054 14. L 3 (2) IN THE FSU, AND L2(2) WHEN .Cf(G, T) IS EMPTY W 0 (R 1 , D) of Din R 1 is normal in both LT and H, so that H::: ...

14.7. FINISHING La(2) WITH (VG1) ABELIAN 1055 contradiction. Hence Po n M = 1, so in particular B f:_ Po. Then as Po contains ea ...

1056 i4. L 3 (2) IN THE FSU, AND L 2 (2) WHEN .Cf(G, T) IS EMPTY PROOF. Let Ko := (KT). By 14.7.30, K 0 LiT E Hz, so without los ...

i4.7. FINISHING L 3 (2) WITH (VG1) ABELIAN 1057 A5 wr Z2 extended by an involution inducing a field automorphism on both K* and ...

1058 14. L 3 (2) IN THE FSU, AND L 2 (2) WHEN .Cr(G, T) IS EMPTY Yo:= (QH,02(Ga)) induces GL(Vy) with kernel 02(Yo) = CqH(Vy)Co ...

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), ...

1060 14. L 3 (2) IN THE FSU, AND L 2 (2) WHEN .Cr(G, T) IS EMPTY Assume (3) fails, and adopt the notation of section B.3 to desc ...

i4.7. FINISHING L 3 (2) WITH (VG1) ABELIAN 106i THEOREM 14.7.40. Assume H E Hz such that H = KLiT for some K E C(H) with K/02,z( ...

i062 i4. L 3 (2) IN THE FSU, AND L 2 (2) WHEN .Cr(G, T) IS EMPTY Pick i to be an H-submodule of UH maximal subject to [UH, K] i. ...

i4.7. FINISHING L 3 (2) WITH (VG1) ABELIAN 1063 over F4, we may apply (1) to Hi to obtain a contradiction. In the second case Ki ...

1064 14. L 3 (2) IN THE FSU, AND L2(2) WHEN .Cf(G, T) IS EMPTY 14.7.40, so the lemma holds. On the other hand, if K/02,z(K) is d ...

i4.7. FINISHING L 3 (2) WITH (VGl) ABELIAN 1065 containing a subgroup isomorphic to A6 or L3(2). Thus either HivI = H4, 3 , or H ...

1066 14. L 3 (2) IN THE FSU, AND L 2 (2) WHEN L:r(G, T) IS EMPTY q(H*, W) :::; 2, and that module is not a strong FF-module. By ...