Absolutely Closed Nil-2 Groups

Using the description of dominions in the variety of nilpotent groups of class at most two, we give a characterization of which groups are absolutely closed in this variety. We use the general result to derive an easier characterization for some subclasses; e.g. an abelian group G is absolutely closed in N2 if and only if G/pG is cyclic for every prime number p . The main result of this paper is a characterization of the absolutely closed groups in the variety N2 (definitions are recalled in Section 1 below). We obtain this result by using the description of dominions in the variety N2 , and applying some ideas D. Saracino used in his classification of the strong amalgamation bases for the same variety [7]. In Section 1 we will recall the main definitions and review the notion of amalgam. In Section 2 we will recall the results of Saracino related to his classification of amalgamation bases of N2 , and we will prove our main result. Finally, in Section 3 we will prove several reduction theorems, and deduce some conditions which are sufficient for a group to be absolutely closed in N2 . We will also give easier to check conditions for special classes of groups; for example, we will show that a finitely generated abelian group is absolutely closed in N2 if and only if it is cyclic. The contents of this paper are part of investigations that developed out of the author’s doctoral dissertation, which was conducted at the University of California at Berkeley, under the direction of Prof. George M. Bergman. It is my very great pleasure to express my deep gratitude and indebtedness to Prof. Bergman, for his advice and encouragement throughout my graduate work and the preparation of a prior version of this paper, and for suggesting Theorem 3.42. Mathematics Subject Classification: 20E06, 20F18 (primary)