From: mareg@lily.csv.warwick.ac.uk (Dr D F Holt) Subject: Re: A question in group theory Date: 5 May 1999 08:05:06 GMT Newsgroups: sci.math Keywords: Groups isomorphic if equal counts of elements of each order? (No) In article <7gnkip\$a51\$1@netnews.upenn.edu>, yclin@hans.math.upenn.edu (Yen-chi R. Lin) writes: >Somebody just asked me about the case when one is abelian but the other >is not. Honestly, I don't know either. > >Cheers, >Roger > >Once upon a time, Yen-chi R. Lin wrote: >> Hi, >> >> Does anybody know (a reference to) a proof or a counterexample to the >> following statement? >> >> >> Let G and H be two finite groups. If the number of the elements of order >> k in G is the same as that in H for every integer k, then G and H are >> isomorphic. >> >> >> I know it is true for abelian groups (just applying the fundamental >> theorem of finitely generated abelian groups.) How about the non-abelian >> group case? Thanks in advance. This is false, and one of the groups could be abelian. The smallest examples have order 16. Let G = C8 x C2 (Cn = cyclic group of order n) and H = < x,y | x^8=1, y^2=1, xy = yx^5 >. In both groups, there are 1,3,4,8 elements of orders 1,2,4,8, respectively. Nearly all conjectures of this kind - if G and H have the same group- theoretical properties then G and H are isomorphic - turn out to be false. Otherwise isomorphism testing would be much easier than it is! Groups of prime power order are often a good place to look for counterexamples, and a brute force computer search is as good a way as any to find them. Derek Holt. ============================================================================== From: Edwin Clark Subject: Re: A question in group theory Date: Tue, 4 May 1999 22:52:26 -0400 Newsgroups: sci.math To: "Yen-chi R. Lin" From MathSciNet: Gnther, Klaus; Lesky, Peter Ein einfaches Isomorphieproblem fr endliche Gruppen. (German. Romanian summary) Bul. Inst. Politehn. Ia\c si (N.S.) 14 (18) 1968 fasc. 3--4 17--20. R. McHaffey [Amer. Math. Monthly 72 (1965), 48--50; MR 30 #1176] proved that a finite abelian group is determined up to isomorphism by its order and the orders of each of its elements. The present authors show that the smallest non-abelian counterexamples have order 16. The Romanian summary quotes McHaffey. ------------------------------------------------------ W. Edwin Clark Department of Mathematics, University of South Florida http://www.math.usf.edu/~eclark/ ------------------------------------------------------