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By Flaass D.G.

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The comments about primitive permu- ments abot~t prime power divisors of (qH - 1) ... (q -1). 1). Using the inductive hypothesis and GO BOUNDS FOR LlNEAI{ GROUPS Sec. 3 Chap. J SOLVABLE LINEAl:. CltOUPS of GL(V) has order 3 ~ the argument in Step 1, we have ' 61 i IV1 3 / 2 I A, and a Hall 3"-~ubgroup has order 24 ::; 92/3 ::; IVI 2/ A. This step is proven. 1) Step 4. We may assume that (i) G if 2 f IGI and q =I- 2 i f(qn)j (ii) n > 3; and that (iii) if q = 2, then n Thus IGI < nqn. To prove (a) and (b), ';'e can· assume that nq ri > q2 n I A and thus 3n > qn ~ 2 n.

Then p and ITI We may assume that conclusion (c) Because 02(G) = 1, we have that p = nand divide 3, whenc~ F( G) is ~xtra-special of order 3 3 . There are limited possibilities for G IC. In each case, p is determined by the fact that (G IC)/( G Ic)' is a p-group. Since p =1= q and r11 I 2n, w~ have one 5,1 SMALL'L1N8AH CROUPS Sec. 2 GIG p qn IUil 2 Z2 Z3 2 3 2 or 3 3 3 2 or 3 3 22 32 2 3 3 3 3 S3 2 4 A4 3 4 S4 2 3 22 or 24 32 5 25 2 no possibility 24 Z5 DlO or F20 ? 5 8 ? 15 yields that GIF:::; f(24).

Hence {Xl - 'YI,Vd generate the same 2-dimensional subspace Proof. 2 (a), we m'ay assume that P is imprimitive. 3, there exists N By :s! P with IP: NI = p such that 1/N = VI EB ... EB Vp for irreducible N ~modules Vi which are transitively permuted by P. Replacing XI Since N IC N(VI by Xl - YI in the above argument, there exists an involution tEN conjugating Xl - ((Xl - YI) +yI)t = YI VI and YI' Now x~t = y~ + (Xl - YI) = Xl. ce generated by {Xl, Yl} thus is = Xl - YI and yr ~ ~), which has order 6.

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2-Local subgroups of Fischer groups by Flaass D.G.


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