Classifying Groups of Certain Orders

Abstract

We will first discuss the question of which integers $n$ have exactly one group of order $n$, namely the cyclic group $\mathbb{Z}/n\mathbb{Z}$. We will see that these are the integers that are relatively prime to the Euler totient function $\phi(n)$. Then we discuss how many groups there are of order $p^3$ for each prime $p$. We end with a couple of interesting results and conjectures pertaining to groups of squarefree order.