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Homenon-isomorphic groups of given order

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# 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

Defines:

Landau's theorem

Related:

BinomialCoefficient, PropertiesOfConjugacy

Type of Math Object:

Theorem

Major Section:

Reference

Parent:

Groups audience:

## Mathematics Subject Classification

20A05*no label found*

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