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.