From: emptyclass@yahoo.com (emptyclass)
Subject: Centralizers in Finite Groups
Date: 26 May 1999 08:55:16 -0400
Newsgroups: sci.math.research
Keywords: groups in which distinct conjugacy classes have distinct sizes

Let G be a finite group, C(g) the centralizer of g in G and 
card(C(g)) its cardinality.  Call G a *-group iff 
card(C(g))=card(C(h)) implies g is conjugate to h in G.  The identity
group of order 1 is a *-group and so is Sym(3)of order 6. Question:
Are there any other finite *-groups? Remarks: A finite *-group has
trivial center and the property that all complex characters are in
fact integral-valued. The only non-trivial finite simple groups whose
complex characters are rational-valued are cyclic of order 2, Sp(6,2)
and O+(8,2), none of which are *-groups.  It follows that a *-group
cannot be a non-trivial finite simple group.  Any comments, hints,
etc. will be appreciated. You may also e-mail responses to 
<emptyclass@yahoo.com>
==============================================================================

From: "A. Caranti" <caranti@science.unitn.it>
Subject: Re: Centralizers in Finite Groups
Date: Thu, 27 May 1999 18:21:27 +0200
Newsgroups: sci.math.research

emptyclass wrote:

> Let G be a finite group, C(g) the centralizer of g in G and
> card(C(g)) its cardinality.  Call G a *-group iff
> card(C(g))=card(C(h)) implies g is conjugate to h in G.  The identity
> group of order 1 is a *-group and so is Sym(3)of order 6. Question:
> Are there any other finite *-groups?

In other words, you are asking whether there are any non-trivial finite
groups, besides Sym(3), in which distinct conjugacy classes have
distinct size.

The answer is definitely no for SOLUBLE groups. This has been settled
in:

J. P. Zhang, Finite groups with many conjugate elements, J. Algebra {\bf
170} (1994), no.~2, 608--624; MR 95i:20028 

and also in:

R. Kn\"orr, W. Lempken and B. Thielcke, The $S\sb 3$-conjecture for
solvable 
groups, Israel J. Math. {\bf 91} (1995), no.~1-3, 61--76; MR
96i:20020 

It seems the problem had been first posed in:

F. M. Markel, Groups with many conjugate elements, J. Algebra {\bf 26}
(1973), 69--74; MR {\bf 48} \#8624 

The MR reviews have some interesting comments and further references on
the story of this problem. It seems progress on this problem for
arbitrary groups is stymied by the fact that the condition does not
behave well with respect to subgroups and quotients.

Andreas Caranti