On ICΦ S – subgroups of finite groups

Abstract

Let H be a subgroup of a group G. Then H is called an IC Φ s − subgroup of G if ( H ∩ [ H , G ] ) H G / H G ⊆ Φ ( H / H G ) H sG / H G , where HsG is the subgroup of H generated by all those subgroups of H which are s-permutable in G. We prove the p-supersolvability of a finite group G under the assumption that certain p-subgroups of G are IC Φ s − subgroups of G. Our main result generalizes and extends some recent work of Gao and Li (2022).