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

Abstract

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 infinitely 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.