Finite Groups with a Prescribed Number of Cyclic Subgroups II

TL;DR

This paper extends the classification of finite groups with a specific number of cyclic subgroups, establishing bounds on group order and enumerating all such groups for certain parameters using computational methods.

Contribution

It proves a universal bound on the order of groups with a given number of cyclic subgroups and enumerates all such groups for small differences using GAP.

Findings

Bound on group order: |G| ≤ 8Δ for groups with |G|-Δ cyclic subgroups.

Finiteness of such groups for each Δ.

Complete enumeration of groups for Δ=1 to 32.

Abstract

T\u{a}rn\u{a}uceanu described the finite groups $G$ having exactly $|G|-1$ cyclic subgroups. In "Finite Groups with a Prescribed Number of Cyclic Subgroups,", we used elementary methods to completely characterize those finite groups $G$ having exactly $|G|-\Delta$ cyclic subgroups for $\Delta=2, 3, 4$ and $5$. In this paper, we prove that for any $\Delta >0$ if $G$ has exactly $|G|-\Delta$ cyclic subgroups, then $|G|\le 8\Delta$ and therefore the number of such $G$ is finite. We then use the computer program GAP to find all $G$ with exactly $|G|-\Delta$ cyclic subgroups for $\Delta=1,\ldots,32$.