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

Anabanti, Chimere and Aroh, A.B. and Hart, Sarah and Oodo, A.R. (2021) A question of Zhou, Shi and Duan on nonpower subgroups of finite groups. Quaestiones Mathematicae , ISSN 1607-3606.

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 in 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 (with k at least 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 greater than or equal to 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.

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