## Publicacions Matemàtiques

- Publ. Mat.
- Volume 54, Number 2 (2010), 317-339.

### Nice Elongations of Primary Abelian Groups

Peter V. Danchev and Patrick W. Keef

#### Abstract

Suppose $N$ is a nice subgroup of the primary abelian group $G$ and $A=G/N$. The paper discusses various contexts in which $G$ satisfying some property implies that $A$ also satisfies the property, or visa versa, especially when $N$ is countable. For example, if $n$ is a positive integer, $G$ has length not exceeding $\omega_1$ and $N$ is countable, then $G$ is $n$-summable if $A$ is $n$-summable. When $A$ is separable and $N$ is countable, we discuss the condition that any such $G$ decomposes into the direct sum of a countable and a separable group, and we show that it is undecidable in ZFC whether this condition implies that $A$ must be a direct sum of cyclics. We also relate these considerations to the study of nice bases for primary abelian groups.

#### Article information

**Source**

Publ. Mat., Volume 54, Number 2 (2010), 317-339.

**Dates**

First available in Project Euclid: 28 June 2010

**Permanent link to this document**

https://projecteuclid.org/euclid.pm/1277731535

**Mathematical Reviews number (MathSciNet)**

MR2675926

**Zentralblatt MATH identifier**

1214.20055

**Subjects**

Primary: 20K10: Torsion groups, primary groups and generalized primary groups

**Keywords**

Nice subgroups elongations abelian groups $\omega_1$-homomorphisms

#### Citation

Danchev, Peter V.; Keef, Patrick W. Nice Elongations of Primary Abelian Groups. Publ. Mat. 54 (2010), no. 2, 317--339. https://projecteuclid.org/euclid.pm/1277731535