Groups with Preassigned Central and Central Quotient Group

Every group G determines two important structural invariants, namely its central C(G) and its central quotient group Q(G) =G/C(G).Concerning these two invariants the following two problems seem to be most elementary.If A and P are two groups, to find necessary and sufficient conditions for the existence of a group G such that A and C(G), B and Q(G) are isomorphic (existence problem) and to find necessary and sufficient conditions for the existence of an isomorphism between any two groups G' and G" such that the groups A, C(G') and C(G") as well as the groups P, Q(G') and Q(G") are isomorphic (uniqueness problem).This paper presents a solution of the existence problem under the hypothesis that P (the presumptive central quotient group) is a direct product of (a finite or infinite number of finite or infinite) cyclic groups whereas a solution of the uniqueness problem is given only under the hypothesis that P is an abelian group with a finite number of generators.There is hardly any previous work concerning these problems.Only those abelian groups with a finite number of generators which are central quotient groups of suitable groups have been characterized before, t 1.Before enunciating our principal results (in 2) some notation and concepts concerning abelian groups will have to be recorded for future reference.If G is any abelian group, then the composition of the elements in G is denoted as addition: x+y.If is any positive integer, then G consists of all the elements x for x in G, and G" consists of all the elements x in G such that x = 0. F(G) is the subgroup of all the elements of finite order in G, that is, the join of the groups G", and F(G, p) is the subgroup of all those elements in F(G) whose order is a power of the prime number p; in other words, F(G, p) is the join of all the groups Gp".It is well known that F(G) is the direct sum of the groups F(G, fi).A set P of elements in G is termed independent, if all the elements in P are non-zero, if none of the elements in B is a multiple of another element in B, and if the group, generated by the elements in P, is the direct sum of the cyclic groups which are generated by the elements in P. If G is generated by