Finite Groups with a Prescribed Number of Cyclic Subgroups

TL;DR

This paper classifies finite groups based on the number of cyclic subgroups they contain, specifically those with a number of cyclic subgroups close to their order, extending previous work for the case of |G|-1.

Contribution

It provides a complete description of finite groups with |G|-Δ cyclic subgroups for Δ=2, 3, 4, and 5, expanding the known classifications.

Findings

Characterization of groups with |G|-2 cyclic subgroups

Classification of groups with |G|-3 cyclic subgroups

Results for groups with |G|-4 and |G|-5 cyclic subgroups

Abstract

Marius T\u{a}rn\u{a}uceanu described the finite groups $G$ having $|G|-1$ cyclic subgroups. We describe the finite groups $G$ having $|G|-\Delta$ cyclic subgroups for $\Delta=2, 3, 4$ and $5$.