Groups with abelian central quotient group

In recent years much progress has been made in the study of groups whose central quotient groups are abelian.1 Such a group is an extension of an abelian group by an abelian group, and the usual method of solving the implied extension problem may be described as follows: If G is such a group, then a maximum abelian subgroup V of G is chosen. V contains the central of G, and G/V represents therefore exactly an abelian group of automorphisms of the abelian group V. Thus it is possible to apply the results of the theory of automorphisms of abelian groups. This method is rather powerful and yields very interesting results. On the other hand it is not restricted to this class of groups and may in fact be applied to all groups with abelian commutator groups. ? Finally it has to be mentioned that this method is not an invariant one, since the maximum abelian subgroups are in no sense uniquely determined. In this paper another method will be indicated. If G is a group whose central quotient group is abelian, then preference is given to a subgroup S which is situated between the central and the commutator group of G. This subgroup S will be left indeterminate as long as possible. But as soon as the final results are reached, either the central or the commutator group of G will take the place of S according to which of these choices will give better results. The extension problem presents itself now in the following form: To characterize those groups whose central contains a given abelian group S and whose quotient group (mod S) is isomorphic to a preassigned abelian group G*. Each group with these properties induces certain invariant relations between the given abelian groups G* and S, and these invariants turn out to be characteristic invariants, provided G* is a direct product of (a finite or infinite number of finite and infinite) cyclic groups. This last hypothesis will be the only restriction of generality imposed on the investigated groups, so that the finite groups with abelian central quotient groups are included in our treatment.