On the intersection of non-normal maximal subgroups of a finite group

Abstract

The subgroup $\Delta (G)$ of a group $G$ is defined to be the intersection of all non-normal maximal subgroups of $G$ (and $\Delta (G)=G$ if all maximal subgroups of G are normal). A group $G$ is called a $T_2$-group if $G/\Delta (G)$ is a $T$-group. Ballester-Bolinches et al. [3] considered the class of $T_2$-groups and gave several results of such groups. In particular, they showed if $G$ is a solvable group, the classes of $T_2$-groups and ${PST}_2$-groups (that is, a group in which $G/\Delta (G)$ is a $PST$-group) are equal. The present work, we introduce the class of $SS{T}_2$-groups which are defined as the groups $G$ for which $G/\Delta (G)$ is an $SST$-group and we show several results of the class $SS{T}_2$-groups. Also, we discuss about equivalency the classes of solvable $PS{T}_2$-groups and solvable $SS{T}_2$-groups.