Theorem. For every positive integer $n$ , there exists only a finite amount of non-isomorphic groups of order $n$ .

This assertion follows from Cayley's theorem, according to which any group of order $n$ is isomorphic with a subgroup of the symmetric group $\mathfrak{S}_n$ . The number of non-isomorphic subgroups of $\mathfrak{S}_n$ cannot be greater than $${n!\!-\!1 \choose n\!-\!1}.$$

The above theorem may be used in proving the following Landau's theorem:

Theorem (Landau). For every positive integer $n$ , there exists only a finite amount of finite non-isomorphic groups which contain exactly $n$ conjugacy classes of elements.

One needs also the

Lemma. If $n \in \mathbb{Z}_+$ and $0 < r \in \mathbb{R}$ , then there is at most a finite amount of the vectors $(m_1,\,m_2,\,\ldots,\,m_n)$ consisting of positive integers such that $$\sum_{j=1}^n\frac{1}{m_j} \;=\; r.$$ The lemma is easily proved by induction on $n$ .