Finite Groups with Given Systems of Conditionally Seminormal Subgroups

The subgroups $$A$$ and $$B$$ are said to be cc-permutable, if $$A$$ permutes with $$B^{x}$$ for some $${x\in\langle A,B\rangle}$$ . A subgroup $$A$$ of a finite group $$G$$ is called conditionally seminormal subgroup of $$G$$ , if there exists a subgroup $$T$$ of $$G$$ such that $$G=AT$$ and $$A$$ is cc-permutable with all subgroups of $$T$$ . In this paper, we proved that saturated formation $$\mathfrak{F}$$ containing the class of all supersoluble groups is closed under product of conditionally seminormal $$\mathfrak{F}$$ -subgroups $$A$$ and $$B$$ with nilpotent derived subgroup or $${(|A|,|B|)=1}$$ . In addition, we studied the structure of a group with given systems of conditionally seminormal subgroups.