Finite Groups in Which Every Proper Characteristic Subgroup is Cyclic

Source abstract

Abstract Let G be a finite, non-cyclic, non-characteristically simple group, such that all its proper characteristic subgroups are cyclic. We call such a group a $$\textrm{CCS}$$ CCS group, short for Characteristic Cyclic Subgroups . In this paper, we provide a complete classification of these groups. As a consequence, we obtain an alternative proof that any skew brace whose multiplicative group is cyclic of p -power order, with p an odd prime, necessarily has a cyclic additive group. Moreover, we describe the multiplicative group of skew braces whose additive group is a solvable, non-nilpotent $$\textrm{CCS}$$ CCS group.