This note explores conditions for dense ICΦ-subgroups in finite groups, showing abelian implications.
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.
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Aseeri et al. (2026) studied this question.
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