1 1 M ar 1 99 8 Z IS AN ABSOLUTELY CLOSED 2 - NIL GROUP

Using the characterization of dominions in the variety of nilpotent groups of class at most two, we prove that the infinite cyclic group is absolutely closed in this variety. We also conclude from the proof that the same is true of arbitrary cyclic groups. The main result of this paper is that cyclic groups are absolutely closed 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 an embedding result which states that any nilpotent group of class at most two may be embedded into a central extension of abelian groups with bilinear cocycle. Since absolutely closed groups are also special amalgamation bases, this is related to a result of Saracino (see [7]) which describes the weak and strong amalgamation bases for N2 . In Section 1 we will recall the main definitions. In Section 2 we will review the notion of amalgam, and recall the result of Saracino already mentioned. We will also show that there are groups which are special amalgamation bases but not weak or strong amalgamation bases in N2 , and we will establish some necessary conditions for a group to be absolutely closed in N2 . Finally, in Section 3 we will prove the main result of this paper, that cyclic groups are also absolutely closed in N2 . The contents of this paper are part of investigations that developed out of the author’s doctoral dissertation, which was conducted 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 this paper; his many suggestions improved the final work in ways too numerous to list explicitly. I also thank Prof. Bergman for allowing me to include the statement and proof of Theorem 3.26, which is due to him and is previously unpublished, and which greatly simplified the argument leading up to Theorem 3.28. Section 1. Preliminaries Recall that Isbell (see [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,