# Cayley’s theorem

Let $G$ be a group, then $G$ is isomorphic^{} to a subgroup^{} of the permutation group^{} ${S}_{G}$

If $G$ is finite and of order $n$, then $G$ is isomorphic to a subgroup of the permutation group ${S}_{n}$

Furthermore, suppose $H$ is a proper subgroup^{} of $G$. Let $X=\{Hg|g\in G\}$ be the set of right cosets^{} in $G$. The map $\theta :G\to {S}_{X}$ given by $\theta (x)(Hg)=Hgx$ is a homomorphism^{}. The kernel is the largest normal subgroup^{} of $H$. We note that $|{S}_{X}|=[G:H]!$. Consequently if $|G|$ doesn’t divide $[G:H]!$ then $\theta $ is not an isomorphism so $H$ contains a non-trivial normal subgroup, namely the kernel of $\theta $.

Title | Cayley’s theorem |
---|---|

Canonical name | CayleysTheorem |

Date of creation | 2013-03-22 12:23:13 |

Last modified on | 2013-03-22 12:23:13 |

Owner | vitriol (148) |

Last modified by | vitriol (148) |

Numerical id | 7 |

Author | vitriol (148) |

Entry type | Theorem |

Classification | msc 20B35 |

Related topic | CayleysTheoremForSemigroups |