non-isomorphic groups of given order

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 ${𝔖}_n$.  The number of non-isomorphic subgroups of ${𝔖}_n$ cannot be greater than

$$\left(\genfrac{}{}{0pt}{}{n!-1}{n-1}\right).$$

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  ,  then there is at most a finite amount of the vectors  $(m_1,m_2,\mathrm{\dots },m_n)$  consisting of positive integers such that

$$\sum _{j=1}^n\frac{1}{m_j}=r.$$

The lemma is easily proved by induction on $n$.