Note on the Number of Finite Groups of a Given Order

Abstract

Let $n$ be a positive integer and $G(n)$ denote the number of non-isomorphic finite groups of order $n$. It is well-known that $G(n)=1$ if and only if $(n,\varphi (n))=1$, where $\varphi (n)$ and $(a,b)$ denote the Euler's totient function and the greatest common divisor of $a$ and $b$, respectively. The aim of this paper is to first present a new proof for the case of $G(n)=2$ and then give a solution to the equation of $G(n)=3$.