Abelian Forcing Sets

Abelian Forcing Sets, by Joseph A. Gallian and Michael Reid
American Mathematical Monthly 100 (1993), no. 6, pp. 580-582.
[DOI] [Math Reviews] [Zentralblatt]
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It is well known that if G is a group in which (xy = 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.
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Updated May 16, 2008.