1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

ia.s. FINISHING THE TREATMENT OF A 6 967 that IVH: VH n Hll::; 2, and [Uk, VH n H^1 ]::; Uk n QH =Uk n V with VUH/UH of rank 1 b ...

968 13. MID-SIZE GROUPS OVER F2 Therefore as IQ : Cq(ui)I = 2, m(Q/Cq(UL)) ::::; 5. Also CL(ui V/V) = LiT, so CLr(ui) is of inde ...

i3.9. CHAPTER APPENDIX: ELIMINATING THE A 10 -CONFIGURATION 969 that u; < v.;. Thus as v.; and u; are normal in CH(Ai)* = CH· ...

970 i3. MID-SIZE GROUPS OVER F2 Hence (3) holds by application of the Burnside Fusion Lemma to the elements of A and B. Next as ...

i3.9. CHAPTER APPENDIX: ELIMINATING THE A 10 -CONFIGURATION 97i orthogonal module for K/B ~ Ot(2) and ltKI = 6. Therefore t^0 nJ ...

972 i3. MID-SIZE GROUPS OVER F2 PROOF. We claim first that Q ::::] Gz. Let Qz := 02(Gz)· By G.2.2 with (z), (t, z), 1 in the rol ...

i3.9. CHAPTER APPENDIX: ELIMINATING THE Aw-CONFIGURATION 973 Next by 13.9.8 and (*), Gz ::::; K and B n Q = (z, tG"). We conclud ...

974 i3. MID-SIZE GROUPS OVER F2 fused under Hi and H 2 , H has one class of involutions. Further Cat (z) = J(T) by 13.9.7, so CH ...

CHAPTER 14 L3(2) ~n the FSU, and L 2 (2) when .Cr(G, T) is empty The previous chapter reduced the treatment of the Fundamental S ...

976 14. L 3 (2) IN THE FSU, AND L 2 (2) WHEN .Cr(G, T) IS EMPTY (a) Mis maximal in M(T) under :S and V = V(M), or (b) V = ((V n ...

14.1. PRELIMINARY RESULTS FOR THE CASE L.:r(G, T) EMPTY 977 LEMMA 14.1.6. (1) M^00 :S CM(V). (2) .C*(G,T) = C(Mc), so that Mc= ! ...

978 14. L 3 (2) IN THE FSU, AND L 2 (2) WHEN .Cr(G, T) IS EMPTY LEMMA 14.1.10. Assume M has a subnormal A3-block X, and 02(M) :: ...

14.1. PRELIMINARY RESULTS FOR THE CASE £.f(G, T) EMPTY 979 Mc = MJ, then Mc is the unique maximal member of M(T) under :S, contr ...

980 14. L 3 (2) IN THE FSU, AND L 2 (2) WHEN .Cr(G, T) IS EMPTY PROOF. SupposeM 1 EM(0 2 ,F•(M)T)andletH:=MnM1. As02,F*(M) _:::: ...

14.2. STARTING THE L 2 (2) CASE OF Cf EMPTY 981 applying 14.1.17 to the preimage Yo in M of O(M), we conclude Y = Yo and Cy(V) : ...

982 14. L 3 (2) IN THE FSU, AND L 2 (2) WHEN L'.f(G, T) IS EMPTY (8) For each HE H*(T, M), H n M is the unique maximal subgroup ...

i4.2. STARTING THE L 2 (2) CASE OF .Cr EMPTY 983 each case, we next define a Bender subgroup Ki of K which, together with Y, wil ...

984 i4. L 3 (2) IN THE FSU, AND L 2 (2) WHEN L.'.r(G, T) IS EMPTY (1) K/0 2 (K) ~ L 3 (4), and some element of T induces a graph ...

i4.2. STARTING THE L 2 (2) CASE OF Cf EMPTY 985 Let Gi := KiS, Gz := BiYS, and Gi,2 := Gin Gz = SBi. Consider the amalgam a:= (G ...

986 i4. L 3 (2) IN THE FSU, AND L 2 (2) WHEN .Cf(G, T) IS EMPTY of two conjugates of Ki, since case (2) of 14.2.8 holds~see e.g. ...