Amalgams and dominions for nilpotent groups of class two
by
Arturo Magidin
Instituto de Matematicas, Universidad Nacional Autonoma de Mexico

A preprint may be found at

Let C be a class of groups. An amalgam of C-groups consists of two groups in C, G and K, and a common subgroup H. The amalgam is weakly embeddable in C if there exists a group M in C, with G and K subgroups of M, and such that H subset G \cap K in M. The amalgam is strongly embeddable if we may choose M with H=G \cap K.

Given a class C, and a group G in C, Isbell defines the dominion of a subgroup H of G in C as the collection of all elements a of G such that any two maps from G to a C-group which agree on H also agree on a.

When the class C is a variety, there is a general algebraic connection between amalgams and dominions, provided by the special amalgams, which are the amalgams where G and K are isomorphic over H. Namely, one may also characterize the dominion of H in G as the smallest subgroup D of G containing H such that the special amalgam of G and G over D is strongly embeddable.

Another related concept is that of amalgamation bases. A group H is a strong (resp. weak) amalgamation base for C if every amalgam of two C-groups G and K over H is strongly (resp. weakly) embeddable in C. A group H is a special amalgamation base if every special amalgam over H is strongly embeddable. It follows from general algebraic facts that a group is a strong amalgamation base for a variety V if and only if it is both a weak and a special amalgamation base for V. The connection of amalgams and dominions also identifies the notion of special amalgamation base with that of being absolutely closed: a group H is absolutely closed in C if and only if for every group G which contains H, the dominion of H in G (in C) equals H.

In 1982, D. Saracino characterized the weak and strong amalgamation bases for the variety of all nilpotent groups of class two. In 1985, Berthold Maier gave necessary and sufficient conditions for weak embeddability of an amalgam in that variety, and in 1986 gave necessary and sufficient conditions for strong embeddability. In 2000, the author gave an explicit description of dominions, and also characterized the special amalgamation bases.

We present a generalization of these results to any subvariety of the variety of nilpotent groups of class two. These classes are known to correspond to pairs of nonnegative integers, (m, n), where n divides n/gcd(m, 2); the pair (m, n) corresponds to the class of all groups satisfying

xm = [x, y]n = [x, y, z] = e.

For example, the characterization of strong embeddability is:

THEOREM. Let G, K, and H be an amalgam in (m, n). The amalgam is strongly embeddable in (m, n) if and only if

1. Gⁿ[G, G] \cap H subset Z(K) and Kⁿ[K, K] \cap H subset Z(G).
2. For every q > 0, q|n, g in G, g' in Gⁿ[G, G], k in K, k' in Kⁿ[K, K], if g^qg', k^qk' in H, then


[gqg', k]=[g, kqk'] in H.

By then looking at dominions, we may use the information they give to compare between embeddability of a single amalgam in different classes.

We also give a characterization of the weak, strong, and special bases in each subvariety, and compare the resulting classes. For example, we obtain the following result:

THEOREM. Let (m, n) subset (m', n').

1. If G in (m, n) is a strong (resp. weak, special) base in (m', n'), then it is also a strong (resp. weak, special) base in (m, n).
2. Every strong (resp. weak) base in (m, n) is also a strong (resp. weak) base in (m', n') if and only if for each prime number p, at least one of the following holds:
   
   (a) ord_p(n) = ord_p(n') = 0.
   (b) ord_p(n)=ord_p(m).
   (c) ord_p(n)=ord_p(n') and ord_p(m) = ord_p(m').
3. Every special base in (m, n) is also a special base in (m', n') if and only if for each prime number p, at least one of the following conditions hold:
   
   (a) ord_p(n), ord_p(n') <= 1.
   (b) ord_p(n)=ord_p(m)=ord_p(n').
   (c) ord_p(n) = ord_p(n') and ord_p(m)=ord_p(m').

Paper reference: arXiv:math.GR/0105233

Date received: March 27, 2002