On the number of cyclic subgroup in finite groups

TL;DR

This paper characterizes finite groups with exactly |G|-3 cyclic subgroups, identifying only the dihedral group D10 and quaternion group Q8 as satisfying this condition.

Contribution

It provides a complete classification of finite groups with a specific number of cyclic subgroups, expanding understanding of subgroup structures.

Findings

G has |G|-3 cyclic subgroups iff G ≅ D10 or Q8

D10 and Q8 are the only groups with this property

The result narrows the possible structures of such groups

Abstract

We study the number of cylic subgroups in finite groups and get that $G$ has $|G|-3$ cyclic subgroups if and only if $G \cong D_{10}$ or $Q_8$.