On the supersolubility of a group with given systems of conditionally seminormal subgroups

Abstract

The subgroups $A$ and $B$ are said to be $\mathrm{cc}$-permutable, if $A$ is permutable with $B^x$ for some ${x\in \langle A,B\rangle}$. A subgroup $A$ of a finite group $G$ is called conditionally seminormal subgroup in $G$, if there exists a subgroup $T$ of $G$ such that $G=AT$ and $A$ is $\mathrm{cc}$-permutable with all subgroups of $T$. In this paper, we proved the supersolubility of a group $G = AB$, where $A$ and $B$ are supersoluble conditionally seminormal subgroups in $G$, in the following cases: the derived subgroup $G^\prime$ is nilpotent; ${(|A|,|B|)=1}$; $G$ is metanilpotent and ${(|G:A|,|G:B|)=1}$; $G$ is metanilpotent and ${(|A/A^{\frak N}|,|B/B^{\frak N}|)=1}$. Besides, we obtained the supersolubility of a group in which maximal subgroups, Sylow subgroups, maximal subgroups of every Sylow subgroup, minimal subgroups, 2‑maximal subgroups are conditionally seminormal subgroups.