What happens if we restrict to an even smaller variety than nil-2 groups, but which still contains non-abelian groups? These varieties are determined by giving two nonnegative integers m and n, such that n divides m if m is odd, and divides (½)m if m is even. The number m is the exponent of the groups, and n the exponent of the commutator subgroups.

   It turns out that one can generalize Maier's result for these classes, if one works with the relatively free groups. The result is a generalization of Maier's result, which in fact also works for varieties of the form (0,n); these are the classes of all nil-2 groups whose commutator is of exponent n; the case n=0 gives the class of all nil-2 groups.

Theorem. Let (A,B;D) be an amalgam of nil-2 groups, and assume that all groups lie in (m,n). The amalgam is strongly embeddable into an (m,n) group if and only if the following two conditions hold:
   (1) The n-th powers times commutators of A which lie in D are central in B. Symmetrically, the n-th power times commutators of B which lie in D are central in A.

   (2) If a lies in A, a' is an n-th power times a commutator in A; b lies in B, b' is an n-th power times a commutator in B; and if a^na' and b^nb' lie in D, then [a^na',b] and [a,b^nb'] are equal and lie in D.

   The case of weak amalgams is similar, though the statement is much more complicated. See my preprint Amalgams of nilpotent groups of class two for the details.

   It is worth noting that embedding in the different (m,n) is not equivalent. Consider the nil-2 group presented by:
       &;     M = <x,y | x^4 = y^4 = [x,y]^4 = [x,y,x] = [x,y,y] = e>.

This group is an (8,4) group (xy is of order 8); let D be the subgroup generated by x; B the group generated by y and x^2; and D their intersection. Both A and B are (4,2) groups; but the amalgam (A,B;D) cannot be embedded into a (4,2) group, even though it can be embedded into an (8,4) group: in a (4,2) group, every square must be central. The generator of D is a square in A, but does not centralize B.

   For weak and strong amalgamation bases we can also generalize Saracino's characterization, and my own for special amalgams. And no group which is not already absolutely closed has an absolute closure. All these results are in the preprint.

   The results have some interesting, and even unexpected, consequences. The example above shows an amalgam (A,B;D) which is strongly embeddable in the class of all nil-2 groups, but not in the variety generated by A and B (i.e. the smallest variety containing them). Also, if we have a group G and a subgroup H, we expect the dominion of H in G to be larger the smaller the category of context. But in the case of the subvarieties of nil-2 groups, it turns out that the dominion in the class of all nil-2 groups is exactly equal to the dominion in the variety generated by G.

    Following the philosophy that dominions tell you how hard it is to upgrade a weak embedding into a strong embedding, the latter result would suggest that it is just as hard to upgrade a weak embedding in a subvariety of nil-2 groups than in the variety of all nil-2 groups. And indeed, if an amalgam is weakly embeddable in the variety corresponding to (m,n), and strongly embeddable in the variety of all nil-2 groups, then it is also strongly embeddable in the variety of all nil-2 groups.