From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Newsgroups: sci.math.research
Subject: Re: group theory problem
Date: 28 Aug 1996 21:16:59 GMT

Ditmar Bachmann wrote:
>	
>Suppose that G is a group and for all a,b from G holds:
>(a*b)^3 = a^3 * b^3
>	 
>Can you derive from this that G is abelian?
>		
>Suppose that G is a group and for all a,b from G holds:
>(a*b)^(-2) = a^(-2) * b^(-2)
>	 
>Can you derive from this that G is abelian?

In article <501upr$ula@newshost.nmt.edu>, Paul Arendt <parendt@nmt.edu> wrote:
>  These are really the same question: multiply the first equation
>by a^(-1) on the left and b^(-1) on the right, and rename a and
>b appropriately.

And the answer to the question(s) is "no". Perhaps the simplest example
would be the non-abelian group of order 27 in which  x^3 = 1  for
every element  x  of the group. (One description of this group is
that it is the set of upper-triangular matrices with  1's  on the 
diagonal, within the group  GL(3, Z/3Z). )

dave

PS -- Am I the only one who feels that some of the recent posts to
sci.math.RESEARCH are potentially homework spoilers? I know the math
newsgroups have had some readjustments in the last year or so -- have
we opened the s.m.r floodgates too wide? I know there is no easy answer,
but this is perhaps the right time of year to discuss this.
Follow-ups set to sci.math (or email as appropriate).