O ct 1 99 8 ABSOLUTELY CLOSED NIL - 2 GROUPS Arturo

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 this to show that a finitely generated abelian group is absolutely closed in N2 if and only if it is cyclic. The main result of this paper is a classification 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 which 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 encouragment throughout my graduate work and the preparation of a prior version of this paper; his many suggestions improved the final work in ways too numerous to list explicitly. Section 1. Preliminaries Recall that Isbell [2] defines for a variety C of algebras (in the sense of Universal Algebra) of a fixed type Ω, and an algebra A ∈ C and subalgebra B of A , the dominion of B in A to be the intersection of all equalizers containing B . Explicitly, domCA(B) = {