A group is called capable if it is a central factor group. In this paper, we establish a necessary condition for a finitely generated non-torsion group of nilpotency class 2 to be capable. Using the classification of two-generator non-torsion groups of nilpotency class 2, we determine which of them are capable and which are not and give a necessary and sufficient condition for a two-generator non-torsion group of class 2 to be capable in terms of the torsion-free rank of its factor commutator group.

In this paper we classify all infinite metacyclic groups up to isomorphism and determine their non-abelian tensor squares. As an application we compute various other functors, among them are the exterior square, the symmetric product, and the second homology group for these groups. We show that an infinite non-abelian metacyclic group is capable if and only if it is isomorphic to the infinite dihedral group

The tensor center of a group $G$ is the set of elements $a$ in $G$ such that $a\otimes g = 1_\otimes$ for all $g$ in $G$. It is a characteristic subgroup of $G$ contained in its center. We introduce tensor analogues of various other subgroups of a group such as centralizers and 2-Engel elements and investigate their embedding in the group as well as interrelationships between those subgroups.

In response to a question posed by P. Erdös, B. H. Neumann showed that in a group with every subset of pairwise noncommuting elements finite there is a bound on the size of these sets. Recently, H. E. Bell, A. A. Klein and the first author showed that a similar result holds for rings. However in the case of semigroups, finiteness of subsets of pairwise noncommuting elements does not assure the existence of a bound for their size. The largest class of semigroups in which we found Neumann's result valid are cancellative semigroups.

In this paper, groups are investigated in which all subgroups, all normal subgroups, or all characteristic subgroups have a proper supplement. This supplement can be either an arbitrary subgroup, a normal or a characteristic subgroup, resulting in nine classes of groups. Properties of these classes are studied such as containment and closure properties, and characterizations for several of these classes are given.

1991 Mathematics Subject Classification Primary 20E34, Secondary 20E15.

The nonabelian tensor square G[otimes ] G of a group G is generated by the symbols g[otimes ] h, g,h ∈ G, subject to the relations $$gg\prime\otimesh=(^gg\prime\otimes^gh)(g\otimesh) and g\otimeshh\prime-(g\otimesh)(^hg\otimes^hh\prime),$$ for all $g,g\prime,h,h\prime \in G< / f>, where $^gg\prime=gg\primeg^{−1}$. The nonabelian tensor square is a special case of the nonabelian tensor product which has its origins in homotopy theory. It was introduced by R. Brown and J.-L. Loday in [4] and [5], extending ideas of J.H.C. Whitehead in [10]. The topic of this paper is the classification of 2-generator 2-groups of class two up to isomorphism and the determination of nonabelian tensor squares for these groups.

According to results of Heineken and Stadelmann, a non-torsion group is a 2-Baer group if and only if it is 2-Engel, and it has all subgroups 2-subnormal if and only if it is nilpotent of class 2. We extend some of these results to values of n greater than 2. Any non-torsion group which is an n-Baer group is an n-Engel group. The converse holds for n = 3, and for all n in the case of metabelian groups. A non-torsion group without involutions having all subgroups 3-subnormal has nilpotency class 4, and this bound is sharp.

Let be a group-theoretic property. We say a group has a finite covering by -subgroups if it is the set-theoretic union of finitely many -subgroups. The topic of this paper is the investigation of groups having a finite covering by nilpotent subgroups, n-abelian subgroups or 2-central subgroups.

R. Baer [12; 4.16] characterized central-by-finite groups as those groups having a finite covering by abelian subgroups. In [6] it was shown that [G: ZC (G)] finite implies the existence of a finite covering by subgroups of nilpotency class c, i.e. ℜc-groups. However, an example of a group is given there which has a finite covering by ℜ2-groups, but Z2(G) does not have finite index in the group. These results raise two questions, on which we will focus our investigations.

Bear's characterisation of central-by-finite groups as groups possessing a finite covering by abelian subgroups if the starting point for this invertigation. We characterise groups with a finite covering by 2-Engel subgroups as groups for which the subgroup of right 2-Engel elements has finite index; and the groups having a finite covering by normal 2-Engel subgroups are exactly the 3-Engel groups among those having a finite covering by 2-Engel subgroups. The second centre of a group having a finite covering by class two subgroups does not necessarily have finite index. However, a group has a finite covering by subgroups in a variety containing all cyclic groups if the margin of thes variety in the group has finite index.