From: rusin@washington.math.niu.edu (Dave Rusin)
Newsgroups: sci.math.research
Subject: Re: number of conjugacy classes in finite groups
Date: 8 Sep 1995 08:13:58 GMT
In article <42n3fh$ij0@nef.ens.fr>,
Philippe Crocy wrote:
>Is it possible to give an explicit upper bound to the order of a finite
>group, depending only on the number of conjugacy classes of this group?
>(that is, to give a function f such that the order of any finite group
>containing x conjugacy classes should be at most f(x) ?)
One upper bound (which I doubt can be sharp) can be shown to exist
using group theory, and perhaps can be made explicit using number theory.
Consider the class equation counting the number of group elements in
each conjugacy class: |G| = Sum( [G:C_G(x_i)], i=1..k ). Dividing by
|G| gives a sum of unit fractions adding to 1:
(*) 1 = Sum( 1/ n_i, i=1..k )
where the n_i = |C(x_i)| may be arranged in increasing order, say.
Now, it is easy to see that for each k, equation (*) has only a
finite number of solutions. (Indeed, n_1 is at most k; then n_2 is
at most (k-1)/(1 - 1/n_1), and in general n_m is at most
(k+1-m)/(1-Sum(1/n_i, i,
Philippe Crocy wrote:
>Is it possible to give an explicit upper bound to the order of a finite
>group, depending only on the number of conjugacy classes of this group?
>(that is, to give a function f such that the order of any finite group
>containing x conjugacy classes should be at most f(x) ?)
Then in article <42ou06$20b@watson.math.niu.edu>, I wrote:
[naive upper bound attained...]
>which then gives an exponential bound, of the form log(f(k)) < a * 2^k.
>
>This seems rather dramatic. I would conjecture that a polynomial bound on
>log(f(k)) exists.
But I knew I had seen this general problem before. A paper by some guy
named Rusin included a reference to
Erdos and Turan, "On some problems of a statistical group theory IV"
Acta Math Acad Sci Hung 19 (1968) 413-435
which shows that the number of conjugacy classes is at least
k >= log log |G|
giving a solution more or less as good as the one I outlined in the previous
article. I guess if Erdos and Turan are willing to give up when they
get this far, then I should too. However, there are papers by Bertam (1974)
and Sherman (1978) which get better mileage by discussing "almost all"
groups, and nilpotent groups, respectively.
dave