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.
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W. Edwin Clark
Department of Mathematics, University of South Florida
http://www.math.usf.edu/~eclark/
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