# 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^{} ${\U0001d516}_{n}$. The number of non-isomorphic subgroups of ${\U0001d516}_{n}$ 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 $n$.

Title | non-isomorphic groups of given order |
---|---|

Canonical name | NonisomorphicGroupsOfGivenOrder |

Date of creation | 2013-03-22 18:56:38 |

Last modified on | 2013-03-22 18:56:38 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 5 |

Author | pahio (2872) |

Entry type | Theorem |

Classification | msc 20A05 |

Related topic | BinomialCoefficient |

Related topic | PropertiesOfConjugacy |

Defines | Landau’s theorem |