Abelian torsion groups having a minimal system of generators

Abstract

Notation and terminology. Let T denote an arbitrary Abelian torsion group. Let G denote an arbitrary primary £-group to be considered fixed in each separate proposition. Let the symbol iff mean and only if, < mean properly contained in, C mean contained in, N\M mean the set of elements in N and not in M, = mean isomorphic to, (Na)aA and if for each aG4 Na is a group let ®aeA Na denote the direct sum of the Na's—© denotes a direct sum. If xEG let h(x) denote the ordinary height of x—see [2]. If 5 is a subset or subgroup of £, let | S denote the power—cardinal number—of 5. Let (S) mean the same thing as 5, and {S} mean the subgroup generated by the elements of 5. If xi, x2, x3, • • • G T, let (xi, x2, x3, • • ■) denote the set whose only elements are Xi, x2, xi, • ■ •, and let {Xi, x2, x3, • • • } = {(xi, Xi, Xi, • • • )}.