Finite groups with semi-subnormal Schmidt subgroups

Abstract

A Schmidt group is a non-nilpotent group in which every proper subgroup is nilpotent. A subgroup \(A\) of a group \(G\) is semi-normal in \(G\) if there exists a subgroup \(B\) of \(G\) such that \(G=AB\) and \(AB_1\) is a proper subgroup of \(G\) for every proper subgroup \(B_1\) of \(B\). If \(A\) is either subnormal in \(G\) or is semi-normal in \(G\), then \(A\) is called a semi-subnormal subgroup of \(G\). In this paper, we establish that a group \(G\) with semi-subnormal Schmidt \(\{2,3\}\)-subgroups is \(3\)-soluble. Moreover, if all 5-closed Schmidt \(\{2,5 \}\)-subgroups are semi-subnormal in \(G\), then \(G\) is soluble. We prove that a group with semi-subnormal Schmidt subgroups is metanilpotent.