On finite groups with given ICΦ-subgroups

A subgroup H of a group G is said to be an IC Phi-subgroup of G if H boolean AND[H, G] <= Phi(H). We analyze the structure of a finite group G under the assumption that some given subgroups of G are IC Phi-subgroups of G. A new characterization of finite abelian groups and some new criteria for 2-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.