Existence of a Unique group of finite order

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. $(n,\varphi (n))=1$, where $\varphi$ 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.