Cut Groups: The Progress And The Problems

TL;DR

This paper reviews the current state of research on cut groups, their generalizations, and discusses both the progress made and the open problems in understanding their algebraic properties.

Contribution

It systematically compiles existing results on cut groups and extends some findings to generalized classes like quadratic rational and semi-rational groups.

Findings

Compiled state-of-the-art results on cut groups.

Extended some results to generalized classes.

Posed open questions for future research.

Abstract

This article focuses on the study of cut groups, i.e., the groups which have only trivial central units in their integral group ring. We provide state of art for cut groups. The results are compiled in a systematic manner and have also been analogously studied for some generalised classes, such as quadratic rational groups and semi-rational groups etc. For these bigger classes, some results have been extended, while others have been posed as questions. The progress and the problems signify the development and the potential that holds in the topic of cut groups and its generalisations.