1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

6.2. IDENTIFYING M22 VIA £ 2 (4) ON THE NATURAL MODULE 687 PROOF. Assume Zs = V n U. We begin by arguing much as at the start of ...

688 6. REDUCING L2(2n) TO n = 2 AND V ORTHOGONAL L x CM ( L) centralizes z and hence lies in H, contrary to 02 ( H n M) = 1. Thu ...

6.2. IDENTIFYING M22 VIA L2(4) ON THE NATURAL MODULE 689 LEMMA 6.2.16. {1) v n Zu = z, so dim(Vu) = 2. (2) (J = (Z1/) and 02(H) ...

69a 6. REDUCING L2(2n) TO n =^2 AND V ORTHOGONAL By definition of the bilinear form on fJ, Z g is the image of Gu (Zs) = W in U, ...

6.2. IDENTIFYING M22 VIA £ 2 (4) ON THE NATURAL MODULE 69i PROOF. By 6.2.13, we may assume Zs < Un V, so the subsequent lemma ...

692 6. REDUCING L2(2n) TO n = 2 AND V ORTHOGONAL deals with the situation where there exists L E £,'j(G, T) with L/0 2 (L) defin ...

Part 3 Modules which are not FF-modules ...

In Part 3, we consider most cases where the Fundamental Setup (3.2.1) holds for a pair L, V such that Vis not a failure of facto ...

CHAPTER 7 Eliminating cases corresponding to no shadow Recall we wish to prove: THEOREM 7.0.1. Assume the Fundamental Setup (3.2 ...

696 7. ELIMINATING CASES CORRESPONDING TO NO SHADOW "{.1. The cases which must be treated in this part Recall we are assuming Hy ...

7.2. PARAMETERS FOR THE REPRESENTATIONS 697 Mv ~ I restr. on n dimV descr. V shadows example U3(2n) n~2 6n natural Sz(2n) odd n ...

698 7. ELIMINATING CASES CORRESPONDING TO NO SHADOW E.3.37; by 7.3.4 this column will give an upper bound on w. Column 6, labele ...

7.4. IMPROVED LOWER BOUNDS FOR r 699 In view of 7.3.2, the column headed m ;?: in Table 7.2.l also provides a lower bound for th ...

700 7. ELIMINATING CASES CORRESPONDING TO NO SHADOW Note that when L ::::) M, Hypothesis 4.4.1 is satisfied by any abelian subgr ...

7.6. FINAL ELIMINATION OF Ls(2) ON 3 EB (^3 701) In the case of A1, we can dig a little deeper to increaser: LEMMA 7.5.5. Lis no ...

702 7. ELIMINATING CASES CORRESPONDING TO NO SHADOW prove that r 2: m; indeed recall that we are postponing that proof that r 2: ...

7.7. MINI-APPENDIX: r > 2 FOR L 3 (2).2 ON 3 EfJ 3 703 Since Zi :S V n V^9 with [V, V^9 ] -:/= 1, 3.2.10.6 says that V i. 02 ...

704 7. ELIMINATING CASES CORRESPONDING TO NO SHADOW 7.7.2. More detailed properties of V 0 and its centralizer. Observe C.M(Vo) ...

7.7. MINI-APPENDIX: r > 2 FOR L 3 (2).2 ON 3 EB 3 705 7. 7.3. Proof of Proposition 7. 7.2. In the remaining two subsections o ...

706 7. ELIMINATING CASES CORRESPONDING TO NO SHADOW Suppose first that (HO) holds. Then W contains the orthogonal sum of the hyp ...