# outer automorphism group

The *outer automorphism group ^{}* of a group is the quotient

^{}(http://planetmath.org/QuotientGroup) of its automorphism group

^{}by its inner automorphism group:

$$\mathrm{Out}(G)=\mathrm{Aut}(G)/\mathrm{Inn}(G).$$ |

There is some variance in terminology about “an outer automorphism.” Some authors define an outer automorphism as any automorphism^{} $\varphi :G\to G$ which is not an inner automorphism. In this way an outer automorphism still permutes the group $G$. However, an equally common definition is to declare an outer automorphism as an element of $\mathrm{Out}(G)$ and consequently the elements are cosets of $\mathrm{Inn}(G)\varphi $, and not a map $\varphi :G\to G$. In this definition it is not generally possible to treat the element as a permutation^{} of $G$. In particular, the outer automorphism group of a general group $G$ does not act on the group $G$ in a natural way. An exception is when $G$ is abelian^{} so that $\mathrm{Inn}(G)=1$; thus, the elements of $\mathrm{Out}(G)$ are canonically identified with those of $\mathrm{Aut}(G)$ so we can speak of the action by outer automorphisms.

Title | outer automorphism group |
---|---|

Canonical name | OuterAutomorphismGroup |

Date of creation | 2013-03-22 14:01:26 |

Last modified on | 2013-03-22 14:01:26 |

Owner | Thomas Heye (1234) |

Last modified by | Thomas Heye (1234) |

Numerical id | 13 |

Author | Thomas Heye (1234) |

Entry type | Definition |

Classification | msc 20F28 |

Related topic | InnerAutomorphism |

Defines | outer automorphism group |