Note on the Number of Finite Groups of a Given Order

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 $(n,\phi(n)) = 1$, where $\phi(n)$ 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 $G(n) = 3$.