American Mathematical Monthly
(1993), no. 6, pp. 580-582.

Abelian Forcing Sets,
by Joseph A. Gallian
and Michael Reid

American Mathematical Monthly
(1993), no. 6, pp. 580-582.

Abstract

It is well known that if *G* is a group in which
(*xy*)² =
*x*²*y*²
for all *x* and *y* in *G* ,
then *G* must be abelian.
It is also true that if (*xy*)^{-1} =
*x*^{-1} *y*^{-1}
for all *x* and *y* in *G* , then again *G*
must be abelian.
We consider groups *G* that satisfy the condition
(*xy*)^{n} =
*x*^{n} y^{n}
for all *x* and *y* in *G* , and all *n* in
a certain set *S* of exponents.
We show that these conditions imply that *G* is abelian,
if and only if the greatest common divisor of the integers
*n*(*n* - 1) ,
as *n* ranges over all elements of *S* , is 2 .
Apparently this was already known,
but we give an entirely elementary proof.

Updated May 16, 2008.