Groups with few nonpower subgroups (Q6564346)

For any positive integer \(r\), the `power' subgroup generated by \(r\)-th powers of elements in any group is a characteristic subgroup. It turns out that there are finite groups which admit characteristic subgroups that are not power subgroups. For cyclic groups, each subgroup is a power subgroup. The numbers \(\operatorname{ps}(G)\) and \(\operatorname{nps}(G)\) of power subgroups and non-power subgroups, respectively, have turned out to be of interest. For finite abelian groups \(G\), it is easy to see that \(\operatorname{ps}(G)\) is the number of divisors of \(\exp(G)\). \textit{C. S. Anabanti} et al. [Quaest. Math. 45, No. 6, 901--910 (2022; Zbl 07557722)] classified the finite groups \(G\) for which, we have \(\operatorname{nps}(G) \leq 4\). There are no \(G\) with \(\operatorname{nps}(G)=1\) or \(2\) and they showed that there are infinitely many groups \(G\) with \(\operatorname{nps}(G)=k\) for ANY \(k \geq 5\). In the paper under review, the author extends these results and classifies all finite groups \(G\) for which \(\operatorname{nps}(G) \leq 9\). For this classification, the author uses results of \textit{C. S. Anabanti} and \textit{S. B. Hart} [Bull. Aust. Math. Soc. 106, No. 2, 315--319 (2022; Zbl 1546.20005)] which were published in the same journal and results of \textit{W. Zhou} et al. [Commun. Algebra 34, No. 12, 4453--4457 (2006; Zbl 1116.20018)].