# faithful group action

Let $A$ be a $G$-set, that is, a set acted upon by a group $G$ with action $\psi :G\times A\to A$. Then for any $g\in G$, the map ${m}_{g}:A\to A$ defined by

$${m}_{g}(x)=\psi (g,x)$$ |

is a permutation of $A$ (in other words, a bijective function from $A$ to itself) and so an element of ${S}_{A}$.
We can even get an homomorphism^{} from $G$ to ${S}_{A}$ by the rule $g\mapsto {m}_{g}$.

If for any pair $g,h\in G$ $g\ne h$ we have
${m}_{g}\ne {m}_{h}$, in other words, the homomorphism $g\to {m}_{g}$ being injective^{}, we say that the action is faithful^{}.

Title | faithful group action |
---|---|

Canonical name | FaithfulGroupAction |

Date of creation | 2013-03-22 14:02:23 |

Last modified on | 2013-03-22 14:02:23 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 8 |

Author | rspuzio (6075) |

Entry type | Definition |

Classification | msc 16W22 |

Classification | msc 20M30 |