When Any Group of N Elements is Cyclic?

TL;DR

This paper provides an accessible proof that a finite group of n elements is cyclic if and only if n and Euler's totient function of n are coprime, requiring only basic number theory knowledge.

Contribution

It offers a simple, elementary proof of a classical result in group theory suitable for students and mathematicians without advanced algebra background.

Findings

Proof confirms the equivalence of cyclicity and coprimality of n and φ(n)

Accessible to students with basic permutations and number theory knowledge

No advanced group theory needed

Abstract

We give a simple proof of the well-known fact: any group of n elements is cyclic if and only if n and \phi(n) are coprime. This note is accessible for students familiar with permutations and basic number theory. No knowledge of abstract group theory is required; a few necessary notions are introduced in the course of the proof. The note could also be an interesting easy reading for mature mathematicians.