On ICΦ-subgroups and p-hypercyclically embedded structure of finite groups

A subgroup [Formula: see text] of a group [Formula: see text] is said to be an [Formula: see text]-subgroup of [Formula: see text] if [Formula: see text]. In this paper, based on research work of Kaspczyk, we analyze [Formula: see text]-hypercyclically embedded structure of a finite group [Formula: see text] under the assumption that some given [Formula: see text]-subgroups of [Formula: see text] are [Formula: see text]-subgroups of [Formula: see text]. Also, all [Formula: see text]-subgroups of symmetric group [Formula: see text] are determined.