1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

13.1. ELIMINATING LE .c;(G,T) WITH L/0 2 (L) NOT QUASISIMPLE 867 LEMMA 13.1.4. Assume LE £+(G, T). Then (1) Either (i) LE £*(G,T ...

868 i3. MID-SIZE GROUPS OVER F2 PROOF. Assume XE B'f_(G,T). Then XE B*(G,T), so Mc:= Nc(X) = !M(XT) by 1.3.7. Also X E B+(G, T), ...

13.1. Eliminating LE .Cf(G, T) with L/0 2 (L) not quasisimple PROOF. By 13.1.4.5, 000 (L) = CL(Vc)· Hence (1) and the statement ...

870 i3. MID-SIZE GROUPS OVER F2 In particular by the claim, LR centralizes Y / 02 (Y), and hence Y normalizes (LR0 2 (Y))^00 = L ...

13.1. ELIMINATING LE .Cr(G, T) WITH L/0 2 (L) NOT QUASISIMPLE 871 K is nontrivial on [DL, R], we may choose A so that KA := [K, ...

872 i3. MID-SIZE GROUPS OVER F2 B.5.1. Now by Theorem B.5.1, either conclusion (3) holds or U E Irr +(L, Ve), in which case B.4. ...

13.1. ELIMINATING LE .C;(G, T) WITH L/0 2 (L) NOT QUASISIMPLE 873 of order 42 with CL(u1)* ~ E4 for u1 E U1. Further either U = ...

874 13. MID-SIZE GROUPS OVER F2 ce~tralizes OP(F(Gu)) and either X = Rp::::; Op( Gu), or X centralizes Op( Gu) and hence F(Gu)· ...

13.1. ELIMINATING LE .L:;(G, T) WITH L/02(L) NOT QUASISIMPLE 875 (i) K =Ko, with k isomorphic to L 3 (p) or (S)L 3 (2n) where 2n ...

876 13. MID-SIZE GROUPS OVER F2 Next assume W 1 (T, U) does not centralize U. Then by the previous paragraph, there is g E G wit ...

i3.2. SOME PRELIMINARY RESULTS ON A5 AND A 6 877 permutation module for L. In particular if L/02(L) = A5, then 02,z(L) s CL(V). ...

878 13. MID-SIZE GROUPS OVER F2 If J(T) ::::;: CT(V) = 02 (LT), then (4) is vacuously true. Thus we may suppose that there is A ...

13.2. SOME PRELIMINARY RESULTS ON A 5 AND A 6 879 PROOF. Part (1) follows from the structure of the A 5 -module. Then by (1), R ...

880 1a. MID-SIZE GROUPS OVER Fz THEOREM 13.2.7. Assume n = 5 and L+/0 2 (L+) ~ A5 for each L+ E .C1(G, T). Then 1-l*(T,M) ~ Ca(Z ...

i3.2. SOME PRELIMINARY RESULTS ON A 5 AND A 6 88i LT by 13.2.4.1, so Na(J(R)):::; M. To complete the proof we show H acts on J(R ...

882 i3. MID-SIZE GROUPS OVER F2 that lemmas= 2 and E = (ei, e 2 ), where ei = ei,2 and e2 = e3,4 are nonsingular. Further T 0 = ...

13.2. SOME PRELIMINARY RESULTS ON A 5 AND A 6 883 Next z = ele2 = el,2e3,4 generates Z, and as K 2 = ((1, 2), (1, 5)), (zK^2 ) = ...

884 13. MID-SIZE GROUPS OVER F2 we may choose D to permute with Ll· Then [D,02(DT)]::::; 031(Ho) nT::::; Rl, so Rl is Sylow in R ...

13.3. STARTING MID-SIZED GROUPS OVER F 2 , AND ELIMINATING U 3 (3) 885 ( 3) There is a T -invariant V K E Irr+ ( K, R2 (KT)) and ...

886 13. MID-SIZE GROUPS OVER F2 (3) The proper overgroups of 'f' in L'f' = Auta(L) are L1'f' and L2T-except when L ~ A5, when on ...