Amalgams of Nilpotent Groups of Class Two

In this paper we will prove analogues of B. Maier’s characterization of weak and strong embeddability of amalgams in N2 [12, 13] for the subvarieties of N2. We will also give analogues of D. Saracino’s characterization of weak and strong amalgamation bases [17], the author’s work on dominions [11] and on amalgamation bases in some varieties of nil-2 groups [8, 9]. Definitions will be recalled below. The results are obtained by extending the methods used by the cited authors. Groups will be written multiplicatively, unless otherwise specified. We will use Z to denote the infinite cyclic group, which we also write multiplicatively. All maps are assumed to be group morphisms unless we explicitly note otherwise. The multiplicative identity of a group G will be denoted by e, and we will use eG if there is danger of ambiguity. For a group G and elements x, y ∈ G, x represents yxy, and the commutator of x and y is [x, y] = xyxy; note that [x, y] = [y, x], and x = x[x, y]. Given subsets A,B of G, not necessarily subgroups, [A,B] denotes the subgroup of G generated by all commutators [a, b] with a ∈ A and b ∈ B. The commutator subgroup of G is the subgroup [G,G], which is also denoted