A Correction to a Result of B. Maier

In a 1985 paper, Berthold J. Maier gave necessary and sufficient conditions for the weak embeddability of amalgams of two nilpotent groups of class two over a common subgroup. Then he derived simpler conditions for some special cases. One of his subsequent results is incorrect, and we provide a counterexample. Finally, we provide a fix for the result. In this note, N2 denotes the variety of nilpotent groups of class two, that is, groups G such that [G, G] ⊆ Z(G). Recall that an amalgam of A and B over the common subgroup D consists of two groups, A and B, and a group D which is a subgroup of both A and B. The amalgam is weakly embeddable in N2 if and only if there is a N2-group G, such that A and B are subgroups of G and D ⊆ A ∩ B (inside G). We then say that G is a (weak) amalgam for A and B over D. If G satisfies the further condition that D = A ∩ B, then G is said to be a strong amalgam. In [1], Berthold J. Maier studied the question of weak embeddability of amalgams in N2. Note that when he says that an amalgam exists, he means that a weak amalgam exists. The Hauptsatz in [1] is a characterization of weak embeddability for amalgams in N2. We quote it here for reference: Theorem 1 (Berthold J. Maier, Haupsatz in [1]). Let A, B ∈ N2, with a common subgroup D ≤ A, B. There exists a weak amalgam of A and B over D in N2 if and only if the following two conditions hold: (1) A2 ∩ D ≤ Z(B) and B2 ∩ D ≤ Z(A). (2) For all k > 0, qi > 0, xi ∈ A and x′i ∈ A2 with x qi i x ′ i ∈ D, yi ∈ B and y ′ i ∈ B2 with yi i y ′ i ∈ D, we have that for every d ∈ D,