1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

15.1. INITIAL REDUCTIONS WHEN Cf(G, T) IS EMPTY 1087 NOTATION 15.1.4. Set M := Mf. We choose V := V(M) in cases (1)-(5) of 15.1. ...

1088 15. THE CASE L'.f(G, T) = 0 that conclusion (6) holds. In case (2) of 15.1.2, conclusion (3) holds. Cases (4) and (5) of 15 ...

15.1. INITIAL REDUCTIONS WHEN .Cf(G, T) IS EMPTY 1089 Since Mis maximal in M(T) under::-.,, we may now apply 14.1.4 to conclude ...

1090 15. THE CASE .C.f(G, T) = 0 for each h E H with [B, Vh] -=f. l; m(A/B) = m(Ui/Cui(A)); and CuH(A) = CuH(B). (3) H/CH(Ui) 9' ...

15.1. INITIAL REDUCTIONS WHEN .Cr(G, T) IS EMPTY 1091 PROOF. As 7t*(T, M) -=!= (i), (1) follows from 15.1.12.4. Next if [V,J 1 ( ...

1092 15. THE CASE .C.r(G, T) =^0 Until the proof of Theorem 15.1.15 is complete, assume G is a counterexample. Thus we are in ca ...

15.1. INITIAL REDUCTIONS WHEN .Cf(G, T) IS EMPTY By (1), 02 (Cc(V(Mc)))::::; 02 (Cc(Z))::::; 02 (Mz), and then by (*) and (**), ...

1094 15. THE CASE .Cr(G, T) =^0 LEMMA 15.1.20. Let LE £(G 1 ,S) with L/0 2 (L) quasisimple and L -f:_ Mz. Set SL:= Sn Land ML:= ...

i5.l. INITIAL REDUCTIONS WHEN .Cr(G, T) IS EMPTY 1095 Now by Remark 0.1.19, we may take C2(SH) = C2(T) and Ci (T) :::; Ci (SH). ...

1096 15. THE CASE .Cf(G, T) =^0 some z E z# induces an inner automorphism on L, then as Mc= !M(Ca(Z)), we have CL(Z(SL)) :::-;; ...

15.1. INITIAL REDUCTIONS WHEN L'.r(G, T) IS EMPTY 1097 2-group, <P(S) induces inner automorphisms on L. Then as S '.::J T by ...

1098 15. THE CASE Cr(G, T) = 0 (ii) 02 (B) = [0^2 (B), J(S)], and either L ~ L 2 (4) or Lis an A5-block. (iii) L ~ U 3 (2n), som ...

15.1. INITIAL REDUCTIONS WHEN .Cr(G, T) IS EMPTY 1099 15.1.17, so by Remark C.1.19 we may take C 2 (S) = C2(T) and C 1 (T) ::::; ...

noo 15. THE CASE .Cr(G, T) = 0 In case (2c), ML is some proper parabolic. In any case, let Po := { (P^8 ) : P is a minimal parab ...

15.1. INITIAL REDUCTIONS WHEN .Cf(G, T) IS EMPTY 1101 In the remainder of the section, let Y := as' ( G 1 nM). As G is a counter ...

1102 15. THE CASE .Cf(G, T) =^0 PROOF. By 15.1.24.2, Lis not an L 3 (2)-block, while in all other cases of 15.1.22, m 3 (L) = 2; ...

i5.l. INITIAL REDUCTIONS WHEN .Cr(G, T) IS EMPTY 1103 (4) UL is a sum of at most n - 1 isomorphic natural modules for L* ~ Ln(2) ...

1104 15. THE CASE .Cr(G, T) = 0 Next 02 (P 0 ) = Cs(V 2 ), and as case (4) of 15.1.7 holds, Cs(Vi) = Cr(V), so Q = 02 (P 0 ). Th ...

15.2. FINISHING THE REDUCTION TO Mf/CMf(V(Mf)) '.:::: o;t(2) 1105 {ii} Case {3} of 15.2.1 holds, Re ~ Z2 inverts 02 (M 0 ) = 02 ...

1106 i5. THE CASE .Cf(G, T) = 0 does not hold, as in that case m 3 (CM(V)) S 1by15.2.3.1. Thus Y = 02 (Mo) ~ Y* by 15.2.3.1, so ...