On 𝑝-abelian groups

Abstract

Finite p-groups in which the relation (xy)P =xPyP is satisfied by all elements x, y have been called p-abelian by C. Hobby [1]. Two examples of classes of such groups are the groups of exponent p and the abelian p-groups. The purpose of this note is to show that these two classes of groups span the class of p-abelian groups in the following sense. If G is a p-abelian group, then there is a finite group P of exponent p, and a finite abelian p-group, A such that G is a factor group of a subgroup of PXA. Since the property of being p-abelian is defined by an identical relation, the class of those groups which satisfy this relation forms a variety. (The p-abelian groups form a proper subset of this variety.) This result can then be restated in varietal terms. We use the notation var G to denote the smallest variety containing the group G. By WUJ3 we denote the smallest variety containing the varieties 21 and e as subclasses.