Finite groups with dense <i>IC</i> Φ-subgroups

A subgroup H of a finite group G is said to be an ICΦ-subgroup of G if H ∩ [H,G] ≤ Φ(H). We say that a finite group has dense ICΦ-subgroups if each non-empty open interval in its subgroup lattice contains an ICΦ-subgroup. In this note, we show that, under a few additional conditions, a finite group with dense ICΦ-subgroups is abelian.