1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

12.3. ELIMINATING A7 807 By 12.2.26, the structure of L, and hence also of Ca(z), are determined, so we can move toward the iden ...

gog 12. LARGER GROUPS OVER F 2 IN .C:j(G, T) Thus we may assume that V is a 4-dimensional irreducible for A1, and it remains to ...

12.3. ELIMINATING A 7 809 {3) Mv ~ 81 or A1, and form= 2, 4, 6, CMv (e(m)) is isomorphic to Z 2 x (^85) or 85; 84 x 83 or a subg ...

810 12. LARGER GROUPS OVER F2 IN .Cj(a, T) Ko = [Ko, J(T)] does not centralize Vo, but 000 (Ko) centralizes Vo by 3.2.14, we con ...

12.3. ELIMINATING A7 811 1, and we compute that this does not hold if A= A 2. This contradiction completes the proof of 12.3.10. ...

812 12. LARGER GROUPS OVER F2 IN Cj(G,T) Let Y := KL2T, U := (ZY), and Y* := Y/Cy(U). As Le = L1L2, Ll < CK(Z), and Z = Cz(L) ...

12.4. SOME FURTHER REDUCTIONS 813 and V/Zv is a natural module for L/0 2 (£). Then as the 1-cohomology of the dual of V/Zv in I. ...

814 12. LARGER GROUPS OVER F2 IN .Cj(G, T) LEMMA 12.4.6. If L ~ L3(2), then Rl E Syb(Gv) and IT: Rll = 2. PROOF. First R 1 E Syl ...

12.4. SOME FURTHER REDUCTIONS 815 LEMMA 12.4.8. r( G, V) > 1. PROOF. Assume that r(G, V) = 1. Then there is a hyperplane U of ...

816 12. LARGER GROUPS OVER F2 IN .Cj(G, T) z inverts Ou 0 := O(Gu 0 ). Similarly z inverts Ou, as we may also apply 1.1.6 and 1. ...

i2.5. ELIMINATING L5(2) ON THE 10-DIMENSIONAL MODULE 8i7 We mention that there is L E .Cj(G,T) with L ~ L 5 (2)/E 210 in the non ...

8i8 i2. LARGER GROUPS OVER F2 IN £.'f (G, T) He, or Ru. To rule out £: 2 (49) or (8)L3(7), observe that in those groups, some el ...

i2.5. ELIMINATING L 5 (2) ON THE 10-DIMENSIONAL MODULE 819 LT ~ (Gi, G3) ~ Na(X), so as M = !M(LT) and 02 (X) =f. 1, Gi ~ M, con ...

820 .12. LARGER GROUPS OVER F2 IN £.j(a,T) by E.6.12; then as U is normal in some Sylow 2-subgroup of LT, E.6.13 supplies a cont ...

i2.5. ELIMINATING L5(2) ON THE iO-DIMENSIONAL MODULE 82i LEMMA 12.5.9. LetTi := Cr(V{), and choose notation with Ti E Syh(CM(V{) ...

S22 i2. LARGER GROUPS OVER F2 IN .Cj(G, T) m 3 (H) = 1, so as n(H) = 2, and a Hall 2'-subgroup of H n Mis a nontrivial 3- group, ...

12.6. ELIMINATING A 8 ON THE PERMUTATION MODULE 823 vectors in V, nondegenerate subspaces of V, etc. For i = 1, 2, 5, let Vi den ...

824 12. LARGER GROUPS OVER F2 IN .Cj(G, T) either Rv = Q or Rv = ((1, 2)). By choice of v, Tv := Cr(v) E Syh(Mv) and IT: Tvl = 4 ...

12.6. ELIMINATING As ON THE PERMUTATION MODULE 825 PROOF. Assume J(Tv) ::::; Q. Then J(Tv) = J(Q), so Nc(Tv)::::; Na(J(Tv)) = Na ...

826 12. LARGER GROUPS OVER F 2 IN .Cj (G, T) whereas (vo) = Gvv (Lv) = Gvv (Kv) with Vv/ (vo) of rank 4 and self-centralizing in ...