1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

i3.3. STARTING MID-SIZED GROUPS OVER F 2 , AND ELIMINATING U 3 (3) 887 PROOF. As KE .C1(G, T), MK= !M(KT) by 13.3.2.2,; then as ...

888 13. MID-SIZE GROUPS OVER F2 We check that Hypothesis C.6.2 is satisfied, with L, B, T, LT, Na(B) in the roles of "L, R, TH, ...

i3.3. STARTING MID-SIZED GROUPS OVER F2, AND ELIMINATING U 3 (3) 889 PROOF. Recall Vis a TI-set in M by 12.2.6, so Hypothesis E. ...

890 i3. MID-SIZE GROUPS OVER F2 Furthermore 031 (H) = K, again using the fact that m 3 (K 2031 (H)) S m3(Na(l)) = Thus Ki SK, s ...

i3.3. STARTING MID-SIZED GROUPS OVER F 2 , AND ELIMINATING U 3 (3) 89i We recall from Notation 13.3.3 that since Cv(L) = 1 by Hy ...

892 13. MID-SIZE GROUPS OVER F2 If 02 (L+) = 1, then 02(L+) s Q 1 s Ca(Vs) by (a), impossible as L+ induces A4 on Vs/V1. (d) 02 ...

13.3. STARTING MID-SIZED GROUPS OVER F2, AND ELIMINATING U 3 (3) 893 the Thompson A x B-Lemma, X is faithful on Cv 3 ( Q J), so ...

894 13. MID-SIZE GROUPS OVER F2 Next Ca(V3) ::; G2 ::; Na(X), so as NL(V3) normalizes Ca(V3), Ca(V3) acts on XNL(Vs). Then as L ...

13.3. STARTING MID-SIZED GROUPS OVER F 2 , AND ELIMINATING U 3 (3) 89S .iii := (r)Ao and Ao are the only such subgroups in case ...

896 13. MID-SIZE GROUPS OVER Fz then as r(G, V) > 3 by 13.3.17.2, Ca(B) S Nc(V^9 ), so that Cv(B) = U. Thus A E ~-k(T, V), so ...

13.4. THE TREATMENT OF THE 5-DIMENSIONAL MODULE FOR A 6 897 LEMMA 13.4.2. (1) M = Na(L) = Na(V) = Oa(Zv). (2) Oa(Z)::::; Oa(ZL): ...

898 13. MID-SIZE GROUPS OVER F2 (3) Irr +(K, R 2 (KT)) ~Irr +(KT, R2(H)), there is VK E Irr +(K, R2(KT), T), and for each such V ...

13.4. THE TREATMENT OF THE 5-DIMENSIONAL MODULE FOR A 6 899 We will now begin to produce subgroups Ho of G which are generated b ...

900 13. MID-SIZE GROUPS OVER F2 Let Ht := H 0 /0 3 1(Ho). As CH 0 (Vo) is a 3'-group, H+ is a quotient of H*. Observe that H+ is ...

i3.4. THE TREATMENT OF THE 5-DIMENSIONAL MODULE FOR A 6 901 As Z = Cz(Hi) x Cz(H2) with Cz(Hi) ~ Z2, there is v E Cz(H 2 ) - Cz( ...

902 13. MID-SIZE GROUPS OVER F2 and Q 1 =J Q 2 , conclusion (2) of F.6.18 holds, and by earlier reduction [Ki, K2] =I-1, so that ...

i3.4. THE TREATMENT OF THE 5-DIMENSIONAL MODULE FOR A 6 903 in case (iv), as there Ji(T) :::) Ho. Hence we are in case (ii), so ...

904 13. MID-SIZE GROUPS OVER Fz of Baum(R 1 ) is normal in Y, and hence not normal in LT as Yi. M = !M(LT). Therefore L is an A ...

13.4. THE TREATMENT OF THE 5-DIMENSIONAL MODULE FOR A 6 905 This establishes the claim that [VH, J(R 1 )] i= 1. By the first par ...

906 13. MID-SIZE GROUPS OVER F2 PROOF. Assume Gz is solvable. Then using B.2.14 as usual, the pair H := Gz, VH := (Z^0 z) satisf ...