It is well known that if G is a group in which
(xy)² =
x²y²
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 =
xn yn
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.