## Abstract

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 = ⟨g^{m} : 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 language | English |
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Pages (from-to) | 901-910 |

Number of pages | 10 |

Journal | Quaestiones Mathematicae |

Volume | 45 |

Issue number | 6 |

DOIs | |

Publication status | Published - 2022 |

## Keywords

- Counting subgroups
- finite groups
- nonpower subgroups

## ASJC Scopus subject areas

- Mathematics (miscellaneous)