Existence of a Unique group of finite order

TL;DR

This paper provides a new proof that the cyclic group of order n is unique among groups of that order if and only if the greatest common divisor of n and Euler's totient of n is 1, using semi-direct products and induction.

Contribution

It introduces an alternative proof of a classical result on the uniqueness of cyclic groups of a given order, utilizing semi-direct products and induction techniques.

Findings

The cyclic group Z_n is unique of order n iff gcd(n, φ(n))=1.

The paper offers a new proof method for the classical result.

The proof employs semi-direct products and induction.

Abstract

Let $n$ be a positive integer. Then cyclic group $Z_n$ of order $n$ is the only group of order $n$ iff g.c.d. $(n,\phi(n))=1$, where $\phi$ denotes the Euler-phi function. In this article we have given another proof of this result using the knowledge of semi direct product and induction.