We investigate the concept of dominion (in the sense of Isbell) in several varieties of nilpotent groups. We obtain a complete description of dominions in the variety of nilpotent groups of class at most two. Then we look at the behavior of dominions of subgroups of groups in N 2 when taken in the context of N c for c > 2 . Finally, we establish the existence of nontrivial dominions in the category of all nilpotent groups.
A group is called capable if it is a central factor group. We consider the capability of nilpotent products of cyclic groups, and obtain a generalization of a theorem of Baer for the small class case. The approach is also used to obtain some recent results on the capability of certain nilpotent groups of class 2. We also establish a necessary condition for the capability of an arbitrary p-group of class k, and some further results.
Using the description of dominions in the variety of nilpotent groups of class at most two, we give a characterization of which groups are absolutely closed in this variety. We use the general result to derive an easier characterization for some subclasses; e.g. an abelian group G is absolutely closed in N2 if and only if G/pG is cyclic for every prime number p .The main result of this paper is a characterization of the absolutely closed groups in the variety N 2 (definitions are recalled in Section 1 below). We obtain this result by using the description of dominions in the variety N 2 , and applying some ideas D. Saracino used in his classification of the strong amalgamation bases for the same variety [7].In Section 1 we will recall the main definitions and review the notion of amalgam.In Section 2 we will recall the results of Saracino related to his classification of amalgamation bases of N 2 , and we will prove our main result. Finally, in Section 3 we will prove several reduction theorems, and deduce some conditions which are sufficient for a group to be absolutely closed in N 2 . We will also give easier to check conditions for special classes of groups; for example, we will show that a finitely generated abelian group is absolutely closed in N 2 if and only if it is cyclic.
Using a new classification of 2-generator p-groups of class 2, we compute various homological functors for these groups. These functors include the nonabelian tensor square, nonabelian exterior square and the Schur multiplier. We also determine which of these groups are capable and which are unicentral.
We give a classification of two-generator p-groups of nilpotency class 2. Using this classification, we give a formula for the number of such groups of order p n in terms of the partitions of n of length 3, and find formulas for the number and size of their conjugacy classes.
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