Finite groups with a certain number of cyclic subgroups

TL;DR

This paper characterizes finite groups with exactly one fewer cyclic subgroup than their order, providing a clear description of such groups and specifically characterizing the symmetric group S_3.

Contribution

It offers a novel characterization of finite groups with |G|-1 cyclic subgroups, including a specific description of S_3.

Findings

Finite groups with |G|-1 cyclic subgroups are characterized.

The symmetric group S_3 is explicitly described within this class.

Abstract

In this short note, we describe the finite groups $G$ having $|G|-1$ cyclic subgroups. This leads to a nice characterization of the symmetric group $S_3$.