On finite groups with given ICΦ-subgroups

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]. We analyze the structure of a finite group [Formula: see text] under the assumption that some given subgroups of [Formula: see text] are [Formula: see text]-subgroups of [Formula: see text]. A new characterization of finite abelian groups and some new criteria for [Formula: see text]-nilpotence and nilpotence of finite groups will be obtained. Moreover, we will obtain two criteria for a finite group to lie in a given solvably saturated formation containing the class of finite supersolvable groups.