Abstract. We show here that the commutator subgroup of a free group of finite rank poses a basis of Bachmuth's type.

Abstract. We answer the question: which property distinguishes the virtually nilpotent groups among the locally graded groups? The common property of each finitely generated group to have a finitely generated commutator subgroup is not sufficient. However, the finitely generated commutator subgroup of F2(var G), a free group of rank 2 in the variety defined by G, is the necessary and sufficient condition.

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If a is an automorphism of order two in an abelian group then each element of the form uu a is a fixed point for a. The question is for which α each fixed point is of the form uu a . There are examples of relatively free groups of rank two where the automorphism σ permuting generators has this property. We describe the automorphisms of order two with all fixed points of the form uu a in a free abelian group of rank two.Let F be a relatively free group of rank two generated by x, y, and σ be the automorphism of F, permuting the generators. If F is abelian, w = x s y t then w = w" implies s = t and w = so each fixed point of σ has a form uu", and hence the group of fixed points is cyclic, generated by χχ σ . It can be deduced from [5] that in a free two-nilpotent group of rank two the group of fixed points for σ is also cyclic, generated by (xy)(xy)

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