A question of Zhou, Shi and Duan on nonpower subgroups of finite groups

C. S. Anabanti*, A. B. Aroh, S. B. Hart, A. R. Oodo

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


A subgroup H of a group G is called a power subgroup of G if there exists a non-negative integer m such that H = ⟨gm : g ∈ G⟩. Any subgroup of G which is not a power subgroup is called a nonpower subgroup of G. Zhou, Shi and Duan, in a 2006 paper, asked whether for every integer k (k ≥ 3), there exist groups possessing exactly k nonpower subgroups. We answer this question in the affirmative by giving an explicit construction that leads to at least one group with exactly k nonpower subgroups, for all k ≥ 3, and in_nitely many such groups when k is composite and greater than 4. Moreover, we describe the number of nonpower subgroups for the cases of elementary abelian groups, dihedral groups, and 2-groups of maximal class.

Original languageEnglish
Pages (from-to)901-910
Number of pages10
JournalQuaestiones Mathematicae
Issue number6
Publication statusPublished - 2022


  • Counting subgroups
  • finite groups
  • nonpower subgroups

ASJC Scopus subject areas

  • Mathematics (miscellaneous)


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