A real chain condition for groups

TL;DR

This paper introduces a weak chain condition for groups related to the absence of certain ordered subsets, and explores its implications for generalized radical groups, extending previous results on stronger chain conditions.

Contribution

It defines a new weak chain condition for groups and studies its effects on generalized radical groups, broadening the understanding of subgroup structures.

Findings

Characterization of groups satisfying the weak chain condition

Extension of previous results on stronger chain conditions

Insights into subgroup properties in radical groups

Abstract

We consider a very weak chain condition for a poset, that is the absence of subsets which are order isomorphic to the set of real numbers in their natural ordering; we study generalised radical groups in which this finiteness condition is set on the poset of subgroups which do not have certain properties which are generalizations of normality. This completes many previous results which considered (apparently) stronger chain conditions.