1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

13.6. FINISHING THE TREATMENT OF A 5 927 Notice cases (c) and (d) of 1.1.5.3 cannot arise: for in those cases z induces an outer ...

928 13. MID-SIZE GROUPS OVER Fz 02(Lv)I = 3, we conclude IB : Be02(B)I = 3 and Be -=f 1. Then Lv/02(Lv) is inverted in Tv n L :: ...

13.6. FINISHING THE TREATMENT OF A 5 929 PROOF. Let U := [z, X] =/:- 1 and suppose that [U, L~] = 1. Then [X, L~] ::; Cx· (U) :: ...

930 13. MID-SIZE GROUPS OVER F2 as B /0 2 (B) is noncyclic of odd order, while 02 (Mv) ~As had cyclic Sylow groups for odd prime ...

i3.6. FINISHING THE TREATMENT OF A 5 93i Ki, Ga in the roles of "X, M". Thus as a E Z(T9) centralizes Ki, 13.6.3.2 applied to Ki ...

932 i3. MID-SIZE GROUPS OVER F2 LEMMA 13.6.21. Ki. M. PROOF. By 13.6.20, K* does not act on L~, so as Lv :::l Mv, 13.6.21 holds. ...

13.6. FINISHING THE TREATMENT OF A 5 933 · Now recall there is a E z^0 n Cv(Lv) with a E Z(Tv) ::; Z(0 2 (KTv)) since F*(KTv) = ...

934 13. MID-SIZE GROUPS OVER F2 LEMMA 13.6.25. (1) Wi(T, V) ::::] LT for i = 0, 1. (2) n(H 1 ) > 1 for each H1 E 1i(T, M). (3 ...

13.7. FINISHING THE TREATMENT OF Aa WHEN (vG1) IS NONABELIAN 935 G1 =KT, or G1 = 0 31 (G1)T with 031 (G1) ~ PGL 3 (4). Hence the ...

936 13. MID-SIZE GROUPS OVER F2 LEMMA 13.7.3. (1) VH :S QH. (2) UH :S Z(VH), so that VH :S Ha. (3) (Ufi) :S 02(LT) = Q. (4) For ...

13.7. FINISHING THE TREATMENT OF A 6 WHEN (VGi) IS NONABELIAN 937 K := 02 (K1), where Ki is a minimal normal subgroup of H* cont ...

938 13. MID-SIZE GROUPS OVER F2 As L 1 centralizes K*, L 1 acts on T n K, so R1 E Syb(KL1R1). Thus if J(R1)::; CR 1 (U), then B ...

i3.7. FINISHING THE TREATMENT OF Aa WHEN (vG1) IS NONABELIAN 939 (a) n = 6, L/02(L) ~ A5, V2 = (ei,2,3,4), and Li has two noncen ...

940 13. MID-SIZE GROUPS OVER F2 So to complete the treatment of the case L 1 :::; K, it remains only to eliminate case (1) with ...

FINISHING THE TREATMENT OF A 6 WHEN (VG1) IS NONABELIAN 941 PROOF. Observe (1) follows from parts (4) and (6) of 13.7.3, a ...

942 13. MID-SIZE GROUPS OVER F2 and DHg ::; Z1 by (5), so that EHg ::; Z1::; VH, and hence EHg ::; Q n He by (2) and 13.7.3.2. B ...

13.7. FINISHING THE TREATMENT OF Aa WHEN (vG1) IS NONABELIAN 943 PROOF. Assume otherwise. Observe there is X1 E 8yl 3 (L 1 ) wit ...

944 i3. MID-SIZE GROUPS OVER F2 on Ui; but this contradicts the structure of FF* -offenders in conclusions (ii)-(iv) of B.5.1.l. ...

13.7. FINISHING THE TREATMENT OF Aa WHEN (vG1) IS NONABELIAN 945 LEMMA 13.7.18. Either (1) K* ~ L2(4) and ·[fK/CuK(K) is the nat ...

946 13. MID-SIZE GROUPS OVER F2 so L 1 has at least six noncentral 2-chief factors. Therefore m(A*) 2 4 by 13.7.10.8. On the oth ...