## Hokkaido Mathematical Journal

### Purifiability in pure subgroups

Takashi OKUYAMA

#### Abstract

Let $G$ be an abelian group. A subgroup $A$ of $G$ is said to be {\it purifiable} in $G$ if, among the pure subgroups of $G$ containing $A$, there exists a minimal one. Suppose that $A$ is purifiable in $G$ and $H$ is a pure subgroup of $G$ containing $A$. Then $A$ need not be purifiable in $H$. In this note, we ask for conditions that guarantee that $A$ is purifiable in the intermediate group $H$. First, we prove that if $A$ is a torsion--free purifiable subgroup of a group $G$ and $H$ is a direct summand of $G$ containing $A$, then $A$ is purifiable in $H$. Next, we characterize the pure subgroups $K$ of a group $G$ with the property that a torsion--free finite rank subgroup $A$ of $K$ is purifiable in $K$ if $A$ is purifiable in $G$.

#### Article information

Source
Hokkaido Math. J., Volume 36, Number 2 (2007), 365-381.

Dates
First available in Project Euclid: 25 June 2010

https://projecteuclid.org/euclid.hokmj/1277472809

Digital Object Identifier
doi:10.14492/hokmj/1277472809

Mathematical Reviews number (MathSciNet)
MR2347431

Zentralblatt MATH identifier
1142.20031

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

#### Citation

OKUYAMA, Takashi. Purifiability in pure subgroups. Hokkaido Math. J. 36 (2007), no. 2, 365--381. doi:10.14492/hokmj/1277472809. https://projecteuclid.org/euclid.hokmj/1277472809