On minimal non-$CL$-groups

TL;DR

This paper investigates the structure of groups that are not $CL$-groups but have all proper subgroups as $CL$-groups, extending recent results in the theory of layered group structures.

Contribution

It extends the understanding of the structure of non-$CL$-groups with all proper subgroups being $CL$-groups, generalizing previous work by Z. Zhang.

Findings

Characterizes the structure of non-$CL$-groups with $CL$-subgroups

Extends recent results on layered group structures

Provides new insights into the influence of layers on group structure

Abstract

If $m$ is a positive integer or infinity, the $m$-layer (or briefly, the layer) of a group $G$ is the subgroup $G_m$ generated by all elements of $G$ of order $m$. This notion goes back to some contributions of Ya.D. Polovickii of almost 60 years ago and is often investigated, because the presence of layers influences the group structure. If $G_m$ is finite for all $m$, $G$ is called $FL$-group (or $FO$-group). A generalization is given by $CL$-groups, that is, groups in which $G_m$ is a Chernikov group for all $m$. By working on the notion of $CL$-group instead of that of $FL$-group, we extend a recent result of Z. Zhang, describing the structure of a group which is not a $CL$-group, but whose proper subgroups are $CL$-groups.