Finite groups with specific number of cyclic subgroups

TL;DR

This paper classifies finite groups with exactly 6, 7, or 8 cyclic subgroups, addressing an open problem and exploring implications for the converse of Lagrange's theorem.

Contribution

It provides a complete classification of finite groups with specific numbers of cyclic subgroups, advancing understanding in group theory.

Findings

Classified all finite groups with 6, 7, or 8 cyclic subgroups.

Connected the classification to the converse of Lagrange's theorem.

Addressed an open problem in the literature.

Abstract

In this note, we classify all finite groups having exactly 6, 7 or 8 cyclic subgroups. This gives a partial answer to the open problem posed by Tarnauceanu (Amer. Math. Monthly, 122 (2015), 275-276). As a consequence of our results, we also obtain an important result concerning with the converse of Lagrange's theorem.