Group Structure from Subgroup and Cyclic Subgroup Counts

Abstract

For a finite group \(G\), let \(\sub(G)\) be the number of subgroups of \(G\), let \(\cyc(G)\) be the number of cyclic subgroups, and let \(π(G)\) be the number of distinct prime divisors of \(|G|\). We study the normalized counts \(λ(G)=\sub(G)/2^{π(G)}\) and \(η(G)=\cyc(G)/2^{π(G)}\). We prove that \(η(G)<5/4\) or \(λ(G)<3/2\) implies that \(G\) is cyclic of squarefree order. The inequalities \(η(G)<2\) and \(λ(G)<5/2\) each force all Sylow subgroups to be cyclic, and hence imply metacyclicity. For the subgroup count, we give an exact arithmetic criterion in the parameters of the corresponding \(ZM\)-presentation. We determine all values with \(1<η(G)<2\) and \(1<λ(G)<5/2\), and prove that \(λ(G)<59/8\) or \(η(G)<4\) implies solvability. Both solvability bounds are sharp. We also describe how cyclic direct factors of coprime order affect the two normalized counts.

57 pages, Comments are welcome