On a product of two formational \(\mathrm{tcc}\)-subgroups

Abstract

A subgroup \(A\) of a group \(G\) is called \(\mathrm{tcc}\)-subgroup in \(G\), if there is a subgroup \(T\) of \(G\) such that \(G=AT\) and  for any \(X\le A\) and \(Y\le T\) there exists an element \(u\in \langle X,Y\rangle \) such that \(XY^u\leq G\). The notation \(H\le G \) means that \(H\) is a subgroup of a group \(G\).  In this paper we consider a group \(G=AB\) such that \(A\) and \(B\) are \(\mathrm{tcc}\)-subgroups in \(G\). We prove that \(G\) belongs to \(\frak F\), when  \(A\) and \(B\) belong to \(\mathfrak{F}\) and  \(\mathfrak{F}\) is a saturated formation of soluble groups such that \(\mathfrak{U} \subseteq \mathfrak{F}\). Here \(\mathfrak{U}\) is the formation of all supersoluble groups.