# Sylow p-subgroup

If $(G,*)$ is a group then any subgroup^{} of order ${p}^{a}$ for any integer a is called a *p-subgroup ^{}*. If $|G|={p}^{a}m$, where $p\nmid m$ then any subgroup $S$ of $G$ with $|S|={p}^{a}$ is a

*Sylow p-subgroup*. We use ${\mathrm{Syl}}_{\mathrm{p}}(G)$ for the set of Sylow p-groups

^{}of $G$.

More generally, if $G$ is any group (not necessarily finite), a Sylow p-subgroup is a maximal $p$-subgroup of $G$.

Title | Sylow p-subgroup |

Canonical name | SylowPsubgroup |

Date of creation | 2013-03-22 12:50:59 |

Last modified on | 2013-03-22 12:50:59 |

Owner | Henry (455) |

Last modified by | Henry (455) |

Numerical id | 8 |

Author | Henry (455) |

Entry type | Definition |

Classification | msc 20D20 |

Synonym | Sylow subgroup |

Synonym | Sylow group |

Related topic | SylowTheorems |

Related topic | ProofOfSylowTheorems |

Related topic | PPrimaryComponent |

Related topic | SylowsThirdTheorem |

Defines | Sylow p-subgroup |

Defines | p-subgroup |