On the Second Cohomology Group of a Finite Group

We shall in fact prove it with C = 2. In the same situation, Aschbacher and Guralnick proved in Theorem A of [1] that \H\G, V)\<\V\. Guralnick has recently improved this bound to \H(G, V)\^\V\l, which is the best possible. At the present time, a proof of the intermediate result \H(G, V)\ «= |V|S is available in preprint form [13]. By using this result, it should be possible with a little extra work to reduce C to 1-5 in Theorem 1, although it is doubtful whether this is worthwhile. Theorem 1 is in any case unsatisfactory, since I know of no examples in which \H(G, V)\>\V\l (we get equality by taking V to be the natural module for G = PSL(2,2"), with n at least 3), and so it may well be true that \H(G, V)\^\V\ for some constant C. There seems to be little prospect of proving such a result at present, however, and Theorem 1 does at least improve upon what appears to be the previous best bound. This has \ep{G){ep{G) + 1) as the exponent of | V|, and it can be derived by observing that a Sylow p-subgroup of G always has a presentation with that many relators. This was obtained and used by P. M. Neumann [20] for the purpose of calculating a bound on the total number of groups of a given order. Indeed, the author hopes to use Theorem 1 in a later paper to derive a bound on the number of perfect groups of a given order. On the subject of presentations, we shall occasionally use the following easy lemma. This might, at some future date, offer the best chance of a significant improvement of Theorem 1, especially if it could be proved that all of the finite simple groups have presentations with small numbers of relations.