Criterions of supersolubility of some finite factorizable groups

Abstract

Let \(A\), \(B\) be subgroups of a group \(G\) and \(\emptyset \ne X \subseteq G\). A subgroup \(A\) is said to be \(X\)-permutable with \(B\) if for some \(x\in X\) we have \(AB^x=B^xA\) [1]. We obtain some new criterions for supersolubility of a  finite group \(G=AB\), where \(A\) and \(B\) are supersoluble groups.  In particular, we prove that a finite group  \(G=AB\) is supersoluble  provided  \(A\), \(B\) are supersolube  subgroups of \(G\) such that  every primary cyclic subgroup of \(A\) \(X\)-permutes with every Sylow subgroup of \(B\) and if in return every primary cyclic subgroup of \(B\) \(X\)-permutes with every Sylow subgroup of \(A\) where \(X=F(G)\) is the Fitting subgroup of \(G\).