Abelian Groups

Date: 09/14/2005 at 21:42:55
From: George
Subject: Abelian groups

If a and b are any elements of a group G and (ab)^3 = a^3*b^3,
is G necessarily Abelian?

I know that (ab)^(-1) = a^(-1)*b^(-1).  We use this repeatedly, and I
think it might be involved in answering this question.

Date: 09/15/2005 at 09:47:47
From: Doctor Vogler
Subject: Re: Abelian groups

Hi George,

Thanks for writing to Dr. Math.  No, it is not necessarily Abelian. 
For example, there is a non-Abelian group of order 27 all of whose
elements have order 3 (except, of course, the identity).

It has a name, in fact, the "Heisenberg group over Z/3Z" and can be
described as the 3-by-3 matrices of the form

  [ 1  a  b ]
  [ 0  1  c ]
  [ 0  0  1 ]

where a, b, and c are elements of the finite field of size 3, namely
Z/3Z.  It can also be written with the presentation

  <x, a, b: x^3 = a^3 = b^3 = 1, ab = ba, xa = abx, xb = bx>

and is, in fact, generated by a and x.

If you have any questions about this or need more help, please write
back and show me what you have been able to do, and I will try to
offer further suggestions.

- Doctor Vogler, The Math Forum
  http://mathforum.org/dr.math/