Nice bases for primary Abelian groups

Abstract

Suppose that A is an Abelian p-group. It is proved that if p ω A is bounded, then A has a bounded nice basis and if p ω A is a direct sum of cyclic groups, then A has a nice basis. In particular, all Abelian p-groups of length < ω.2 along with all simply presented Abelian p-groups are equipped with bounded nice bases. It is also shown that if length(A)≤ ω.2 and A/p ω A is countable, then A possesses a bounded nice basis as well as if length(A)≤ ω.2 and p ω A is countable, then A possesses a nice basis. Moreover, contrasting with these claims, we demonstrate that if length(A)=ω.2 and A/p ω A is torsion-complete with finite Ulm-Kaplansky invariants, then A does not have a bounded nice basis. If in addition p ω A is torsion-complete, then A does not have a nice basis, respectively. Finally, we construct a summable $p^{\omega+2}$ -projective group (thus a summable group with a nice basis) which is not a direct sum of countable groups. This answers in negative our question posed in (Atti Sem. Mat. Fis. Univ. Modena e Reggio Emilia, 2005). Keywords: Bounded nice basis, Nice basis, Bounded groups, Direct sums of cyclic groups, Summable groups, $p^{\omega+n}$ -projective groups, Simply presented groups, Σ-groups, Torsion-complete groups, Large subgroups, Countable extensions, Bounded extensions Mathematics Subject Classification: 20K10, 20K15