# 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 $\mathfrak{S}_{n}$.  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,  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 $n$.

Title non-isomorphic groups of given order NonisomorphicGroupsOfGivenOrder 2013-03-22 18:56:38 2013-03-22 18:56:38 pahio (2872) pahio (2872) 5 pahio (2872) Theorem msc 20A05 BinomialCoefficient PropertiesOfConjugacy Landau’s theorem