From: mareg@csv.warwick.ac.uk (Dr D F Holt) Newsgroups: sci.math Subject: Re: Groups and my software Date: 17 Oct 1997 09:56:13 +0100 In article <61vvts\$34r\$1@gannett.math.niu.edu>, rusin@vesuvius.math.niu.edu (Dave Rusin) writes: > >Wilbert Dijkhof asked for a reference >for this result I had posted: > Let a_n be the number of non-isomorphic groups of order p^n. > Then log_p (a_n) is asymptotic to (2/27)n^3. > >Graham Higman showed (Proc London Math Soc 10 (1960) 24-30) that >the number b_n = log_p (a_n) / n^3 satisfies > 2/27 - o(n) <= b_n <= 2/15 + o(n) >where the implied constant is independent of p. Later, Charles Sims >(Proc London Math Soc 15 (1965) 151-166) improved the upper bound to > b_n <= 2/27 + o(n) >(Again this is independent of p, although Higman's lower bound is >a little tighter than Sims' upper bound.) I am a little confused by your o(n) terms, since you have already divided through by n^3. More precisely, Higman proved n^3b_n >= 2n^3/27 + O(n^2) and Sims proved n^3b_n <= 2n^3/27 + O(n^(8/3)) > >Many more delicate results are also known for p-groups, partitioning the >set of those groups in various ways and then enumerating the parts. Counting >non-p-groups is rather more problematic, but good results have been obtained >by Laszlo Pyber (Ann Math 137 (1993) 203-220). Yes, Pyber showed that for any m>0, if f(m) denotes the number of isomorphism classes of groups of order m, and x is the highest exponent of any prime dividing m, then 2 (2/27 + o(1))x f(m) <= m . Note that the exponent agrees asymptotically to Sims' when m is a prime power. So, at least as far as the exponent is concerned, this result is getting close to best possible. He does this by showing that compared to the number of p-groups themselves, the number of groups with an assigned collection of isomorphism classes of Sylow p-subgroups is relatively small. This result depends on the classification of finite simple groups, in so far as it assumes an upper bound for the number of finite simple groups of any given order. (Using the classification, this upper bound is 2.) It is interesting to note that Higman's lower bound is derived by considering special p-groups only. (That is, groups G in which the centre and derived group are the same, and both Z(G) and G/Z(G) are abelian of exponent p.) So, it is a reasonable conjecture that in some sense almost all groups are special 2-groups, although the results proved to date are nothing like strong enough to prove that. By the way, I believe that the smallest m for which f(m) is not currently known is 192 (unless this has changed recently). However, this does not mean that 192 presents insuperable difficulties - rather nobody to date has had the time or energy to carry out a complete enumeration. Derek Holt.