## Hokkaido Mathematical Journal

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

### Purifiability in pure subgroups

#### 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

**Permanent link to this document**

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

**Keywords**

purifiable subgroup pure hull strongly ADE decomposable group mixed basic subgroup

#### 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