1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

15.3. THE ELIMINATION OF Mr/CMr(V(Mr)) =Sa wr Z 2 1147 PROOF. First U is nonabelian by 15.3.52, so that Z = cl>(U) by 15.3.48 ...

1148 15. THE CASE .Cf(G, T) =^0 2-chief factors, sos+ 2=r+1 by(*). Further m(U/X) = 1 and m(X/E) = r, so using (2), · m(E) = m(U ...

15.3. THE ELIMINATION OF Mr/CMr(V(Mr)) = S 3 wr Z 2 1149 PROOF. Observe first that s > 0: For if s = 0, then by 15.3.54.4, Y ...

1150 15. THE CASE .Cf(G, T) = (/) and K = 02 (K), we conclude using Coprime Action that K centralizes Qc. Thus Qc = CT(TK)· By 1 ...

15.3. THE BLIMINATION OF Mf/CMr(V(Mr)) = Ss wr Z 2 1151 Let Bi:= Gp(Di)V. Then Bi~ E54 and I has four orbits on Bf: zI and vr, a ...

1152 15. THE CASE .Cf(G, T) = 0 Assume HE 'H*(T, M) with n(H) > 1. Then in view of 15.3.2.6, His described in E.2.2. In parti ...

15.3. THE ELIMINATION OF Mf/CMr(V(Mr)) = S 3 wr Z 2 1153 (4) GE/QE ~ S3 x S3 and XE= [XE, J(R)]. PROOF. By 15.3.5.2, if A E AtT) ...

1154 15. THE CASE .Cr(G, T) = f/J where N := m(OvE(B)/OvE(A)). Therefore m(A) ::'.'.'. m(B) + N and hence 2m(A) ::'.'.'. 2m(B) + ...

i5.4. COMPLETING THE PROOF OF THE MAIN THEOREM 1155 By 15.3.67.2, J(TE) ~ R, so that J(TE) :Si YFTE, and so Yp acts on X. As Xis ...

1156 15. THE CASE .Cr(G, T) = (/J Because of Hypothesis 15.4.1, members of t,(G, T) have few overgroups, so that t,*(G, T) is no ...

15.4. COMPLETING THE PROOF OF THE MAIN THEOREM 1157 {1) 3!M(XT), so that XE C(G, T). {2) X:::; 02,p(M) for ME M(XT). In particul ...

1158 15. THE CASE .Cr(G, T) =^0 LEMMA 15.4.6. If Mis maximal in M(T) with respect to :S and [V(M), J(T)] = 1, then M is the uniq ...

15.4. COMPLETING THE PROOF OF THE MAIN THEOREM 1159 We come to the main result of this subsection: THEOREM 15.4.8. Cc(Z) = T. Th ...

1160 15. THE CASE .Cr(G, T) = 0 15.4.10, X::; CM;(V(Mi)) =:Hi, so as Hi::; Ca(Z) and X :::1 Ca(Z), X :::1 Hi. Thus 02 (X) ::; 02 ...

15.4. COMPLETING THE PROOF OF THE MAIN THEOREM 1161 By 15.4.11.1, 02(Yo) = Cy 0 (V(M)). IfT acts irreducibly onX, then Yo lies i ...

1162 15. THE CASE .Cr(G, T) = r/J PROOF. Pick YE Y n M, and apply Theorem 3.1.1 to S, Na(S), YT in the roles of "R, M 0 , H". No ...

15.4. COMPLETING THE PROOF OF THE MAIN THEOREM 1163 F*(C1(z)) = 02(C1(z)) for each z E z#, and m(Z) = 2by15.4.14.2, we conclude ...

1164 i5. THE CASE .C.r(G, T) = r/J Set K := (Xr^0 ) and recall KE 3(G, T). As Mi= Nc(X), Xis not normal in Io since Io i. Mi by ...

i5.4. COMPLETING THE PROOF OF THE MAIN THEOREM 1165 By the claim, J+ is described in Theorem F.6.18. Therefore as Vr is an FF- m ...

1166 15. THE CASE .Cr(G, T) = 0 Although by this point it may feel like something of an anticlimax, we have also completed the p ...