Cyclic non-$S$-permutable subgroups and non-normal maximal subgroups

Abstract

A finite group $G$ is said to be a $T$-group (resp. $PT$-group, $PST$-group) if normality (resp. permutability, $S$-permutability) is a transitive relation. Ballester-Bolinches et al. gave some new characterizations of the soluble $T$-, $PT$- and $PST$-groups. A finite group $G$ is called a $T_c$-group (resp. ${PT}_c$-group, ${PST}_c$-group) if each cyclic subnormal subgroup is normal (resp. permutable, $S$-permutable) in $G$. The present work defines the ${NNM}_c$-, ${PNM}_c$-, and ${SNM}_c$-groups and presents new characterizations of the wider classes of soluble $T_c$-, ${PT}_c$-, and ${PST}_c$-groups.