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.

摘要

如果存在一个非负整数m使得H = ⟨g ^m : g ∈ G⟩，则群G的子群H称为 G 的幂子群。G的任何不是幂子群的子群称为G的非幂子群。Zhou、Shi 和 Duan 在 2006 年的一篇论文中询问对于每个整数k ( k ≥ 3) 是否存在恰好拥有k的群非权力子群。我们通过给出一个明确的构造来肯定地回答这个问题，该构造导致至少一个具有恰好k个非幂子群的群，对于所有k ≥ 3，并且当k是复合且大于 4时，有无数这样的群。此外，我们描述了对于基本阿贝尔群、二面体群和最大类 2 群的情况，非幂子群的数量。