From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Subject: Re: Conjugacy in a non-Abelian group
Date: 2 May 2001 15:29:50 GMT
Newsgroups: sci.math
Summary: Maximum (relative) number of conjugacy classes in a non-abelian group?

In article <k5pea0syib3y@forum.mathforum.com>,
Tim Brooks <timbrooks@REMOVEmy-deja.com> wrote:
>If G is a finite non-Abelian group with n elements how large can be
>the number of conjugacy classes in G ?

At most (5/8) |G|.

An elementary counting exercise shows that if two elements of a finite
group are chosen at random (with replacement, uniform probability) then
the probability the two elements commute is  k/|G|,  where  k  is (as
always!) the number of conjugacy classes. So you are essentially asking, 
"What is the probability that two elements of a finite group commute?",
which happens to be the title of a paper: Pacific J. Math. 82 (1979), 237-247

dave