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 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).