1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

7.7. MINI-APPENDIX: r > 2 FOR Ls(2).2 ON 3 E!l S 707 This completes the proof of Lemma 7.7.6 and Proposition 7.7.2. 7.7.4. Pr ...

708 7. ELIMINATING CASES CORRESPONDING TO NO SHADOW LEMMA 7.7.9. (1) If case (a) of G.2. 7.3 holds, then K is an A1-block. (2) [ ...

7.7. MINI-APPENDIX: r > 2 FOR L 3 (2).2 ON 3 E9 3 709 We now eliminate the cases (a)-(d), (e) with K* ~ A 6 , and (f); in all ...

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CHAPTER 8 Eliminating shadows and characterizing the J 4 example We begin by reviewing the cases remaining after the work of the ...

71Z 8. ELIMINATING SHADOWS AND CHARACTERIZING THE J4 EXAMPLE which are not SQTK-groups; as a consequence we obtain an improved b ...

8.1. ELIMINATING SHADOWS OF THE FISCHER GROUPS 713 Thus Hypothesis C.2.8 holds, and we may apply Theorem C.4.8. If Ca(x) i M, th ...

714 8. ELIMINATING SHADOWS AND CHARACTERIZING THE J4 EXAMPLE If m(.A) = 5, then by H.14.3.1, A= KQ· Then W = V by H.15.4.4, cont ...

8.2. DETERMINING LOCAL SUBGROUPS, AND IDENTIFYING J 4 715 the hypotheses of 3.1.9 hold with LT in the role of "M 0 ": Recall we ...

716 8. ELIMINATING SHADOWS AND CHARACTERIZING THE .14 EXAMPLE m(V/Z 1 ) = 3k = 6n and m(.A) = 4n. Therefore m(Z1) = m(V) - 6n = ...

8.2. DETERMINING LOCAL SUBGROUPS, AND IDENTIFYING J 4 717 are as described in the Table, then m(U) = m(Zr ), so Zr= U. Further t ...

718 8. ELIMINATING SHADOWS AND CHARACTERIZING THE J4 EXAMPLE subgroup I of H are solvable, so that 1 = n(I) = k by E.1.13; hence ...

8.2. DETERMINING LOCAL SUBGROUPS, AND IDENTIFYING J 4 719 If j is L2(2), then conclusion (a) of (3) holds for k = 1, and Pis the ...

720 8. ELIMINATING SHADOWS AND CHARACTERIZING THE J4 EXAMPLE PROOF. We begin with the proof of (1), although we will obtain (3) ...

8.2. DETERMINING LOCAL SUBGROUPS, AND IDENTIFYING J 4 721 If 02 (H+) is of order 3 or 5, then H =IT, so that (1) holds. Thus we ...

722 8. ELIMINATING SHADOWS AND CHARACTERIZING THE J4 EXAMPLE Let Lz := CL(z)^00 and C := C/Z. LEMMA 8.2.11. (1) Lz E C(CM(z)). ( ...

8.3. ELIMINATING La(2) 12 ON 9 723 L;/02(L;) ~ A5 that the A5-module V/Vz does not arise in A.3.14.^3 That is, (1) holds. By 8.2 ...

724 8. ELIMINATING SHADOWS AND CHARACTERIZING THE J4 EXAMPLE REMARK 8.3.4. The second case of lemma 8.3.3 in fact arises in the ...

8.3. ELIMINATING L 3 (2) 12 ON 9 725 We claim Y:::;; M. IfY is solvable, then n(Y) = 1 by E.1.13, so Y:::;; M by 8.3.6. So suppo ...

726 8. ELIMINATING SHADOWS AND CHARACTERIZING THE J4 EXAMPLE PROOF. We first prove (1). Assume the first statement in (1) fails. ...