On the number of cyclic subgroups of a finite group. (English) Zbl 1499.20036
Summary: Let \(G\) be a finite group and let \(c(G)\) be the number of cyclic subgroups of \(G\). We study the function \(\alpha(G)=c(G)/|G|\). We explore its basic properties and we point out a connection with the probability of commutation. For many families \(\mathcal{F}\) of groups we characterize the groups \(G\in\mathcal{F}\) for which \(\alpha(G)\) is maximal and we classify the groups \(G\) for which \(\alpha(G)>3/4\). We also study the number of cyclic subgroups of a direct power of a given group deducing an asymptotic result and we characterize the equality \(\alpha(G)=\alpha(G/N)\) when \(G/N\) is a symmetric group.
MSC:
| 20D10 | Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks |
| 20B35 | Subgroups of symmetric groups |
| 20D30 | Series and lattices of subgroups |
Software:
GAPReferences:
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