1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

i5.2. FINISHING THE REDUCTION TO Mr/CMf(V(Mf)) ~ Oj:"(2) 1107 Suppose Li is not quasisimple. Then by 1.2.1.4, fA := F(Li) = F*(L ...

l108 is. THE CASE ..Cf(G, T) = 0 We next construct a subgroup Yi of H with Yi of order 3, Yi :::l Mi, and Mi = (Min Mc) Yi. Supp ...

15.2. FINISHING THE REDUCTION TO Mr/CMr(V(Mr)) c:: o:t(2) 1109 (1) Y1Rc/02(Y1Rc) ~ S3, D10, or 8z(2). (2) Cy 1 (V) = 02(Y1). (3) ...

1110 i5. THE CASE .Cf(G, T) = 0 M = !M(YiT) by 15.2.6.3, so that Oz((YiT,H)) = 1, and hence part (e) holds. To verify part (d), ...

i5.2. FINISHING THE REDUCTION TO Mr/CMf(V(Mf)) c:= 0!(2) 1111 PROOF. Recall B = B(A) for some A E .6.'(V), and we may assume wit ...

1112 15. THE CASE .Cf(G, T) =^0 holds. Then we compute that X acts faithfully on [V, BJ =: Zs, so X :::; Na (Zs) :S Mc by 15.2.1 ...

15.2. FINISHING THE REDUCTION TO Mr/CMf(V(M£)) ~ Of(2) 1113 LEMMA 15.2.18. (1) SE Syb(I). (2) B := Baum(T) = Baum(S) and O(I, B) ...

11i4 i5. THE CASE .Cf(G, T) = 0 LEMMA 15.2.21. (1) If S is not irreducible on K/02(K) then KS = HiH2, K1 = K2 fort ET-S, and Kc ...

15.2. FINISHING THE REDUCTION TO Mr/CMf(V(Mf)) ~ Oj"(2) 1115 and [K1, K2] S Kin K2· Now S acts on X, F*(I) = 02 (J) by hypothesi ...

1116 i5. THE CASE .C.r(G, T) =^0 Xis described in C.1.34, then Xis an L 3 (2)-block. This completes our preliminary reductions. ...

15.2. FINISHING THE REDUCTION TO Mr/CMf(V(Mf)) ~ o:t(2) 1117 (a) ·u is the sum of isomorphic natural modules, and M 1 is an end- ...

1118 15. THE CASE .Cr(G, T) = (/J If L ~ L 2 (p) for p a Mersenne or Fermat prime, then p > 7 by 15.2.24, and CL 0 (Z) =SL. T ...

15.2. FINISHING THE REDUCTION TO Mf/CMf(V(M£)) ~ o:t(2) 1119 PROOF. We claim that 03 ,(Mc) :::;: Mn Mc: for otherwise we may cho ...

1120 15. THE CASE .Cr(G, T) =^0 respectively. For a:= "(1g and /3 := 'YoY, set Ua := UfI, Za := z~, and V,a :=VY. Let b := b(r, ...

15.3. THE ELIMINATION OF Mf/CMf(V(Mf)) = Ss wr Z2 15.3.1. Preliminary results. Recall by Hypothesis 14.1.5.2 that Mc= !M(Ca(Z)). ...

1122 15. THE CASE .Cr(G, T) = 0 Thus M = (M1 x M2)(f';, where Mi:= CM(V3-i), Vi= [V, Mi], Mi~ o;-(Vi) ~ 83, and tis an involutio ...

15.3. THE ELIMINATION OF Mr/CMf(V(Mf)) =Sa wr Z2 1123 LEMMA 15.3.9. V < 02(Y), so that Y ~ A4 x A4. PROOF. Assume V = 02 (Y), ...

1124 15. THE CASE .Cr(G, T) = 0 PROOF. Let SS T1 E Syl2(I). By 15.3.6.2, Na(S) SM, so as IT: SI = 2, T 1 SM. Thus either (1) hol ...

15.3. THE ELIMINATION OF Mr/CMr(V(Mr)) = 83 wr Z 2 1125 (2) Assume that m3(L) = 1 and Y+ induces inner automorphisms of L/0 2 (L ...

1126 15. THE CASE .Cr(G, T) = 0 LEMMA 15.3.16. {1) WL E R 2 (LR) n R 2 (LRY+) and L+ is quasisimple. {2) m 3 (L) ~ 1, YL i= l, R ...