Self-Small Abelian Groups

Abstract

This paper investigates self-small abelian groups of finite torsion-free rank. We obtain a new characterization of infinite self-small groups. In addition, self-small groups of torsion-free rank 1 and their finite direct sums are discussed.2000 Mathematics subject classification: primary 20K21.

Warfield dualities induced by self-small mixed groups

“…Proposition 2.11 (see [1]). Let A ¼ AðT; RÞ be a standard mixed self-small group of torsion-free rank 1 and n a positive integer.…”

“…A group is called a standard mixed selfsmall group of torsion-free rank 1 if it is isomorphic to a group AðT; RÞ. It is proved in [1] that the construction of AðT; RÞ is independent, up to isomorphism, of the choice of the generators a p : Proposition 2.8 (see [1]). The set of pairs ðT; RÞ, where T is a torsion group with cyclic p-components and R is a subgroup of Q which is p-divisible for all p A SðTÞ, is a complete set of independent invariants for standard mixed self-small groups of torsion-free rank 1.…”

“…Moreover, every self-small group of torsion-free rank 1 with infinite torsion subgroup can be described using standard mixed self-small groups of torsion-free rank 1: Theorem 2.9 (see [1]). A group A is self-small of torsion-free rank 1 if and only if A G B l R with B a finite group and 0 0 R c Q or A G B l AðT; RÞ for some finite group B and some standard mixed self-small group of torsion-free rank 1.…”

“…Moreover, in general A 1 l A 2 is not a self-small group (see [1]). In the present paper we are interested only in a particular case.…”

“…The proof of the following lemma is a simple exercise. To conclude this section we recall from [1] two results concerning finite index subgroups of finite powers A n , where A is a self-small group of torsion-free rank 1. Quasi-isomorphic torsion groups are defined in [6, p. 11] and characterized in [6, Exercise 1.10].…”