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 ============================================================================== From: "A. Caranti" 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