# $p$-subgroup

Let $G$ be a finite group^{} with order $n$, and let $p$ be a prime integer.
We can write $n={p}^{k}m$ for some $k,m$ integers, such that $k$ and $m$ are coprimes^{} (that is, ${p}^{k}$ is the highest power of $p$ that divides $n$).
Any subgroup^{} of $G$ whose order is ${p}^{k}$ is called a Sylow $p$-subgroup.

While there is no reason for Sylow $p$-subgroups to exist for any finite group, the fact is that all groups have Sylow $p$-subgroups for every prime $p$ that divides $|G|$. This statement is the First Sylow theorem^{}

When $|G|={p}^{k}$ we simply say that $G$ is a $p$-group.

Title | $p$-subgroup |
---|---|

Canonical name | Psubgroup |

Date of creation | 2013-03-22 14:02:14 |

Last modified on | 2013-03-22 14:02:14 |

Owner | drini (3) |

Last modified by | drini (3) |

Numerical id | 8 |

Author | drini (3) |

Entry type | Definition |

Classification | msc 20D20 |

Related topic | PGroup4 |

Defines | Sylow $p$-subgroup |

Defines | Sylow p-subgroup^{} |

Defines | $p$-group |

Defines | p-group^{} |