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  <s9400842@mail.student.utwente.nl> 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.