1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

13.4. THE TREATMENT OF THE 5-DIMENSIONAL MODULE FOR A 6 907 holds by A.1.7.2. Finally consider any e E Ov(d) - {d}. Then e E Oa( ...

908 13. MID-SIZE GROUPS OVER F2 02 (CK(Z)) :S: M by 13.4.2.2. Then LK :S: B(M) = L by 13.4.2.5, so that LK :S: 02 (CL(Z)) :S: 02 ...

13.4. THE TREATMENT OF THE 5-DIMENSIONAL MODULE FOR A 6 909 As L/02(L) ~ A6 by (3), R = Ri. Also either U E Irr +(K, R 2 (KT)), ...

910 lS. MID-SIZE GROUPS OVER F2 normalize each other. Thus L 2 centralizes U 2 := (Z{f^2 ), and K2 centralizes U1 := (zL^2 ). Bu ...

13.4. THE TREATMENT OF THE 5-DIMENSIONAL MODULE FOR A 6 911 that H2,o/02(H2,o) ~ L3(2), and hence 02,3(H 2 ,o) = 02 (H 2 , 0 ) i ...

gi2 i3. MID-SIZE GROUPS OVER F2 PROOF. Assume [V, Vt] # 1. By hypothesis Vt S Rz and V S Rg, so Vt and V normalize each other. L ...

13.4. THE TREATMENT OF THE 5-DIMENSIONAL MODULE FOR A 6 913 FF-modules by Theorem B.5.1, so by 13.4.14, we are in case (i) of 13 ...

914 13. MID-SIZE GROUPS OVER F2 section including Definition F.7.2, where in particular "(o, "(1 are the cosets G1, G2. For 'Y = ...

13.5 .. THE TREATMENT OF A5 AND Aa WHEN (v;^1 ) IS NONABELIAN 915 We will now obtain a contradiction to our assumption that H is ...

916 13. MID-SIZE GROUPS OVER F2 PROOF. As in the proof of 13.4.5, this follows from 13.3.2, once we observe that by 13.3.2, we m ...

13.5. THE TREATMENT OF A5 AND Aa WHEN (v;^1 ) IS NONABELIAN 917 LEMMA 13.5.6. Assume n = 6; then (1) V::; 02(G2). (2JVPnV=Vl. (3 ...

918 13. MID-SIZE GROUPS OVER F2 It remains to treat the case where [V3, VB] = V 1 = [V/, V]. Here m(VB / Cvg (V)) = 1, so V indu ...

i3.S. THE TREATMENT OF As AND Aa WHEN (v;^1 ) IS NONABELIAN. 9i9 In the remainder of the section, H will denote any member of 'H ...

920 i3. MID-SIZE GROUPS OVER F2 {6) H = Gi and Gi is the unique member of Hz. (1) M =LT and V = 02(M). {8) If n = 5 then QH =UH, ...

i3.5. THE TREATMENT OF A5 AND A 6 WHEN (v;^1 ) IS NONABELIAN 92i V x D, we conclude Q E !32(KDD), and in particular Q contains 0 ...

922 i3. MID-SIZE GROUPS OVER F2 Next as y E L 2 - H, z -=f-zY E Vi::::; QH::::; Ca(j) so that j E Ca(zY) =HY, and similarly jY E ...

i3.5. THE TREATMENT OF A5 AND Aa WHEN (v;^1 ) IS NONABELIAN 923 LEMMA 13.5.20. V* centralizes F(H*). PROOF. If [Op(H), V] =/= 1 ...

924 13. MID-SIZE GROUPS OVER F2 By (iii), m([UH, VJ)= 2, so as m(V) = 1, q(KV*,UH)::::; 2. Therefore B.4.2 and B.4.5 describe K ...

13.5. THE TREATMENT OF A5 AND Aa WHEN (v;^1 ) IS NONABELIAN 925 LEMMA 13.5.23. K* is not As. PROOF. Assume K* ~As. Then by 13.5. ...

926 i3. MID-SIZE GROUPS OVER F2 13.6. Finishing the treatment of A In this section, we complete the treatment of the case in the ...