Finite Non-abelian Simple Groups Which Contain a Non-trivial Semipermutable Subgroup

Abstract

A subgroup H of a group G is called semipermutable if it is permutable with every subgroup K of G with (|H|,|K|)=1, and s-semipermutable if it is permutable with every Sylow p-subgroup of G with (p,|H|)=1. In this paper, we classify all finite non-abelian simple groups which contain a non-trivial semipermutable (s-semipermutable) subgroup. As a corollary of our main result, we give a complete answer to an unsolved problem in group theory proposed by V.S. Monakhov in 1990.