Finite groups in which every proper characteristic subgroup is cyclic

TL;DR

This paper classifies finite groups where every proper characteristic subgroup is cyclic, revealing their structure and implications for related algebraic objects like skew braces.

Contribution

It provides a complete classification of finite CCS groups, advancing understanding of their structure and applications.

Findings

Characterization of all finite CCS groups

Connection to minimal non-cyclic skew braces

Structural properties of these groups

Abstract

Let $G$ be a finite non-cyclic, non-characteristically simple group with the property that all proper characteristic subgroups of $G$ are cyclic. We call such a group $\mathrm{CCS}$ group, short for \emph{Characteristic Cyclic}. In this paper, we provide a complete classification of these groups. As an application of our main result, we also make some progress toward the classification of minimal non-cyclic skew braces.