On a finite group with OS-propermutable Sylow subgroup

Abstract

A Schmidt group is a non-nilpotent group whose every proper subgroup is nilpotent. A subgroup A of a group G is called OS-propermutablein G if there is a subgroup B such that \(G = NG(A)B\), where AB is a subgroup of G and A permutes with all Schmidt subgroups of B. We proved \(p\)-solubility of a group in which a Sylow \(p\)-subgroup is OS-propermutable, where \(p\geq 7\) 7. For \(p < 7\) all non-Abelian composition factors of such group are listed.