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 AB1 is a proper subgroup of G for every proper subgroup B1 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.