## Hokkaido Mathematical Journal

### On purifiable torsion-free rank-one subgroups

Takashi OKUYAMA

#### Abstract

First, we give a necessary and sufficient condition for a torsion-free rank-one subgroup of an arbitrary abelian group to be purifiable in a given group and show that all pure hulls of a purifiable torsion-free rank-one subgroup are isomorphic. Next, we show that if a $T(G)$-high subgroup A of an abelian group $G$ is purifiable in $G$, then there exists a subgroup $T'$ of $T(G)$ such that $G=H\oplus T'$ for every pure hull $H$ of $A$ in $G$. An abelian group $G$ is said to be a strongly ADE decomposable group if there exists a purifiable $T(G)$-high subgroup of $G$. We present an example $G$ such that not all $T(G)$-high subgroups of a strongly ADE decomposable group G are purifiable in $G$. Moreover, we characterize the strongly ADE decomposable groups of torsion-free rank 1. Finally, we use previous results to give a necessary and sufficient condition for an abelian group of torsion-free rank 1 to be splitting.

#### Article information

Source
Hokkaido Math. J., Volume 30, Number 2 (2001), 373-404.

Dates
First available in Project Euclid: 22 October 2012

https://projecteuclid.org/euclid.hokmj/1350911959

Digital Object Identifier
doi:10.14492/hokmj/1350911959

Mathematical Reviews number (MathSciNet)
MR1844825

Zentralblatt MATH identifier
0991.20041

Subjects
Primary: 20K21: Mixed groups
Secondary: 20K27: Subgroups

#### Citation

OKUYAMA, Takashi. On purifiable torsion-free rank-one subgroups. Hokkaido Math. J. 30 (2001), no. 2, 373--404. doi:10.14492/hokmj/1350911959. https://projecteuclid.org/euclid.hokmj/1350911959