1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

i3.8. FINISHING THE TREATMENT OF A 6 947 Recall we may choose 9b with ('Yo, 'Yi)gb = bb-i, "!)· Then U'"Y := Ufl, V'"Y := VJ/, Q ...

948 i3. MID-SIZE GROUPS OVER F2 If 02 (K) = L 0 , then H ::::; Na(0^2 (K)) = N 0 (L 0 ) ::::; M by 13.2.2.9, contrary to Hf:. M. ...

13.8. FINISHING THE TREATMENT OF A 6 949 Suppose one of the first three cases holds, namely UH is an irreducible module. To elim ...

950 13. MID-SIZE GROUPS OVER F2 So assume that [F, U'Y] = [F, V'Y]. Then [F, V'Y] s [F, U'Y]Vi s U'YV1 as UH acts on U'Y. If V 1 ...

i3.8. FINISHING THE TREATMENT OF A 6 95i a contradiction. Thus P i M, so by minimality of H, H = PLiT and LiT is irreducible on ...

952 13. MID-SIZE GROUPS OVER F2 By 13.8.13, H is nonsolvable, so there exists K E C(H). By 13.8.14, F(H*) = Z(0^2 (H*)), so K* i ...

13.8. FINISHING THE TREATMENT OF A 6 953 u;, E Q(H*,UH), so in particular u;, acts quadratically on UH, and hence it fol- lows f ...

954 i3. MID-SIZE GROUPS OVER F2 PROOF. Assume otherwise. By 13.8.16, we may take H = KLiT. By 13.8.17, K* ~ L 2 (2n), (8)L3(2n), ...

13.8. FINISHING THE TREATMENT OF Aa 955 As n is odd, B =Be, so B centralizes V 3. Therefore as C1H (B) = 0, we conclude V3 i. IH ...

956 i3. MID-SIZE GROUPS OVER F2 PROOF. We begin with the proof of (2); as usual, we may take H = KLiT. By 13.8.17 and 13.8.20, K ...

13.8. FINISHING THE TREATMENT OF A 6 957 Thus L/02(L) ~ A5, so that Lo and Li,+ = X are the two T-invariant sub- groups of 3-ran ...

958 i3. MID-SIZE GROUPS OVER F2 LEMMA 13.8.23. Assume m(U;) = 1, and K is nontrivial on W. Then (1) u'Y induces transvections on ...

i3.8. FINISHING THE TREATMENT OF A 6 959 and H* ~ 87, B.4.2 and B.4.5 say that UH is either a natural module or the sum of a 4-d ...

960 13. MID-SIZE GROUPS OVER F2 Thus m(U H) = 6. Since the case with a single nontrivial 2-chief factor which is an A 8 -module ...

i3.8. FINISHING THE TREATMENT OF A 6 96i We claim UH has a unique maximal submodule W. Assume not; then (writing J(UH) for the J ...

962 i3. MID-SIZE GROUPS OVER Fz by 13.8.18.4. Also by 13.8.23.1, u; induces transvections on wand UH, so H has a unique noncentr ...

i3.8. FINISHING THE TREATMENT OF A 6 963 (a) L/02(L) 9:! A5, H 9:! A6 or 86! and UH is the natural module for K on which Li has ...

964 13. MID-SIZE GROUPS OVER F2 LEMMA 13.8.29. {1) K2 E C(G2) with K2/02(K2) ~ L3(2). (2) G2 = K2L2,+T and Ch= k2 x j2 ~ L3(2) x ...

i3.8. FINISHING THE TREATMENT OF Aa 965 u; = UJr :S: VH. Thus as VH is abelian, VH centralizes u;, and hence also u'Y. Therefore ...

966 i3. MID-SIZE GROUPS OVER F2 CQH (Vi), we conclude that He s CQH (Vi) = Q n QH S Q. Thus He centralizes V, and hence He also ...