On generating systems for abelian groups

Abstract

The purpose of this note is to investigate two kinds of generating sets for an abelian group. It is well known that if an abelian group has a projective cover then it has a very special kind of generating set: a free basis. The idea of a projective cover of an abelian group is dual to that of a divisible hull when the divisible hull is defined as an essential injective extension. This note considers the dualization of two other equivalent definitions of a divisible hull. This yields the definitions of a q-cover and an i-cover. It is shown that an abelian group has a q-cover if and only if it has a minimum system of generators (in the sense of Khabbaz). It is also shown that the only torsion free abelian groups which have q-covers are the free groups and that the torsion subgroup of an abelian group has a q-cover if the group does. These two facts give rise to the suspicion that there is a direct connection between groups with q-covers and splitting groups; two examples are given which show that this suspicion is false. Finally it is shown that very many abelian groups have i-covers; included among these are all torsion free groups and all groups which have q-covers. Throughout this note all groups are abelian and the notation, for the most part, follows [1]. In particular: r(G) is the rank of G, r*(G) is the reduced rank of G, Gt is the torsion subgroup of G, Z is the additive group of integers and Zn is the integers mod n. A subgroup H of G is small in G if for any subgroup K of G, { H, K } = G implies K = G. DEFINITION (KHABBAZ [3 ]). A subset S of a group G is a minimum system of generators (abbreviated: M.s.g.) for G if {S} =G and no finite subset Si of S may be replaced by a smaller subset T of G in such a way that { (S-Si), T } = G. DEFINITION. A q-cover for a group G is a free group F together with an epimorphism 0 of F onto G such that for any proper direct summand F1 of F, 0 [F1 ] $ G. An i*-cover of G is a free group F together with an epimorphism 4 of F onto G such that if F1 is free, i1 an epimorphism of F1 onto G and ais a homomorphism of F into F1 such that 6Vo=4 then ois a monomorphism.