1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

14.2. STARTING THE L 2 (2) CASE OF .Cf EMPTY 987 as J2 has two classes of involutions, cc;,(t) is isomorphic to a Sylow 2-subgro ...

9SS 14. L 3 (2) IN THE FSU, AND L2(2) WHEN L.:r(G, T) IS EMPTY We prove: THEOREM 14.2.20. Let HE 1-i*(T,M). Then (1) If H/0 2 (H ...

14.3. FIRST STEPS; REDUCING (vG1) NONABELIAN TO EXTRASPECIAL 989 Thus it remains to deal with the case where a is the^2 F 4 (2)- ...

990 14. L 3 (2) IN THE FSU, AND L 2 (2) WHEN .Cf(G, T) IS EMPTY Thus in this section, and indeed for the remainder of the chapte ...

i4.3. FIRST STEPS; REDUCING (VG1) NONABELIAN TO EXTRASPECIAL 99i LEMMA 14.3.5. Assume L/02(L) ~ L 2 (2)' and HE H(T) with IH: Tl ...

992 14. L 3 (2) IN THE FSU, AND L 2 (2) WHEN .Cr(G, T) IS EMPTY Z 3 x Z 63. By 14.3.5, any subgroup of order 3 or 5 permuting wi ...

14.3. FIRST STEPS; REDUCING (VG1) NONABELIAN TO EXTRASPECIAL 993 rank 1. Thus for either choice of i = 1, 2, there exists 9i E L ...

994 14. L 3 (2) IN THE FSU, AND L2(2) WHEN .Cr(G, T) IS EMPTY U = VZu, contradicting U nonabelian. Thus L/0 2 (L) ~ L2(2)'. Here ...

14.3. FIRST STEPS; REDUCING (VG1) NONABELIAN TO EXTRASPECIAL 995 Next by 12.8.13.3, z& 2'! Zu, so z& -=I-1. Let K := 02 ...

996 14. L 3 (2) IN THE FSU, AND L 2 (2) WHEN .Cr(G, T) IS EMPTY U; hence E has rank 1, and so V = E. On the other hand, U^9 nQ:: ...

i4.3. FIRST STEPS; REDUCING (vG1) NONABELIAN TO EXTRASPECIAL 997 By 14.3.18.4, A :'::) hQ, so that as L = 02 (! 2 ), K ::::; Na( ...

998 14. L 3 (2) IN THE FSU, AND L 2 (2) WHEN .Lf(G, T) IS EMPTY 02 (Gd) ::::; 8d, and Ai = 02 (Kd8d), so we conclude that (d) = ...

14.3. FIRST STEPS; REDUCING (V^0 1) NONABELIAN TO EXTRASPECIAL 999 Let G1 := GA/A. From the structure of Aut(K) in 14.3.19.10, s ...

iooo i4. L 3 (2) IN THE FSU, AND L 2 (2) WHEN .Cr(G, T) IS EMPTY (i) if~ 83 x 83 , with Li :'SI if. Further if Ziji-=/:- 1 then ...

14.3. FIRST STEPS; REDUCING (VGi) NONABELIAN TO EXTRASPECIAL 1001 LEMMA 14.3.25. Z(LT) n U = 1. PROOF. Assume ZL := Z(LT) n U I ...

1002 14. L 3 (2) IN THE FSU, AND L 2 (2) WHEN £,f(G, T) IS EMPTY from section 12.8 with Vi in the role of "Vi". Similarly V1 Vf ...

14.3. FIRST STEPS; REDUCING (vG1) NONABELIAN TO EXTRASPECIAL 1003 K1 is G2(2)' or A5, then Ki contains all elements of order 3 i ...

1004 14. Ls(2) IN THE FSU, AND L 2 (2) WHEN £,f(G, T) IS EMPTY Suppose next that case (i) of 14.3.23.2 does not hold; we will el ...

i4.4. FINISHING THE TREATMENT OF (VG1) NONABELIAN ioo5 Let Xi be of order 3 in Li. Then Q = [Q, Xi]CQ(Xi) with [Q, Xi] = [U, Xi] ...

1006 14. L 3 (2) IN THE FSU, AND L 2 (2) WHEN Cf(G, T) IS EMPTY Theorem 14.3.26 handled the case where U is nonabelian but not e ...