1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

i4.6. ELIMINATING L 2 (2) WHEN (vG1) IS ABELIAN This contradiction establishes the claim that D = 1. Now by (e) and (g): (h) 02( ...

1028 14. L 3 (2) IN THE FSU, AND L 2 (2) WHEN .Cr(G, T) IS EMPTY Set I*:= I/0 2 (I). Then U]r :::1 TJ, while m(UH/CuH(QH)) 2 4 b ...

14.6. ELIMINATING L 2 (2) WHEN (VG1) IS ABELIAN 1029 In the remaining conclusions of F.6.18, there is K E C(I) with K ::::] I, a ...

1030 14. L 3 (2) IN THE FSU, AND L2(2) WHEN .Cf(G, T) IS EMPTY Thus T 1 =Tu by 14.6.6.1. Therefore by the hypothesis of part (5) ...

14.6. ELIMINATING L 2 (2) WHEN (Val) IS ABELIAN 1031 normal subgroup of H*. Thus Tis maximal in PT= H, and P ~ Zp or EP2, since ...

i032 i4. L 3 (2) IN THE FSU, AND L 2 (2) WHEN .Cr(G, T) IS EMPTY vectors of UK. Then S := Cs(zY) for some Sylow 2-subgroup S of ...

i4.6. ELIMINATING L 2 (2) WHEN (vGi) IS ABELIAN 1033 order divisible by each prime dividing 2m - 1, and one of these primes is l ...

i034 i4. L 3 (2) IN THE FSU, AND L 2 (2) WHEN .Cr(G, T) IS EMPTY now apply F.9.18 to Ki, Gi in the roles of "K,H": As the ot(16) ...

i4.6. ELIMINATING L2(2) WHEN (VG1) IS ABELIAN 1035 Kz = KiK2, we obtain J = [J, Ki]. As m5(G2):::; 2 and m 5 (K 2 ) = 1, m 5 (J) ...

1036 i4. L 3 (2) IN THE FSU, AND L 2 (2) WHEN .Cr(G, T) IS EMPTY [K,ui] =/= 1 =/= [K,uiui]. Further in all cases of 14.6.14, KS ...

14.6. ELIMINATING L 2 (2) WHEN (v^0 1) IS ABELIAN 1037 satisfies the hypotheses of 14.6.10.5, so m((V^12 )) = 3 by that lemma, c ...

1038 14. L 3 (2) IN THE FSU, AND L 2 (2) WHEN L:r(G, T) IS EMPTY (3) H^00 = KK+ with K, K+ normal C-components of H, and K/02(K) ...

14.6. ELIMINATING L 2 (2) WHEN (vG1) IS ABELIAN 1039 (4) Ca(u) EI, so that I* is nonempty. PROOF. Set To := CT(u). To prove (1) ...

1040 14. L 3 (2) IN THE FSU, AND L 2 (2) WHEN .Cr(G, T) IS EMPTY where the last inequality follows from the fact that Uy induces ...

14.6. ELIMINATING L2(2) WHEN (vG1) rs ABELIAN 1041 least 3. However K+ = Cc((z,zg})^00 is invariant under SE Syb(Cc((z,zg}), so ...

1042 14. Ls(2) IN THE FSU, AND L 2 (2) WHEN .Cr(G, T) IS EMPTY We now derive a contradiction, hence showing that no examples sat ...

14.7. FINISHING L 3 (2) WITH (VG1) ABELIAN Often we can show that D 7 < U 7 , and in those situations we also adopt: NOTATION ...

1044 i4. L 3 (2) IN THE FSU, AND L 2 (2) WHEN .Cf(G, T) IS EMPTY PROOF. As Li ::::I H, H normalizes 02 (LiQH) =Li. Then 14.5.15. ...

i4.7. FINISHING L 3 (2) WITH (vG1) ABELIAN 1045 previous paragraph. Therefore as JTA : TiJ ::::; JT: TiJ = 2 = JTA : 02(LoTA)J, ...

i046 i4. L 3 (2) IN THE FSU, AND L 2 (2) WHEN .Cf(G, T) IS EMPTY PROOF. Assume otherwise and let Hi := LiT and Hz the minimal pa ...