On a theorem of Stelzer for some classes of mixed groups

TL;DR

This paper explores classes of mixed groups with the cancellation property, showing their Walk-endomorphism rings have the unit lifting property, especially for certain torsion-free groups of rank at most 4.

Contribution

It identifies specific classes of mixed groups where the cancellation property implies the Walk-endomorphism ring has the unit lifting property, extending Stelzer's theorem.

Findings

Groups with cancellation property have Walk-endomorphism rings with the unit lifting property.

Self-small torsion-free groups of rank ≤ 4 decompose into free and special parts.

Decomposition results apply to classes of mixed groups with cancellation property.

Abstract

We identify some classes $\mathcal{C}$ of mixed groups such that if $G\in \mathcal{C}$ has the cancellation property then the Walk-endomorphism ring of $G$ has the unit lifting property. In particular, if $G$ is a self-small group of torsion-free rank at most $4$ with the cancellation property then it has a decomposition $G=F\oplus H$ such that $F$ is free and the Walk-endomorphism ring of $H$ has the unit lifting property.