# inner automorphism

Let $G$ be a group. For every $x\in G$, we define a mapping

$${\varphi}_{x}:G\to G,y\mapsto xy{x}^{-1},y\in G,$$ |

called conjugation^{} by $x$.
It is easy to show the conjugation map is in fact, a group automorphism^{}.

An automorphism^{} of $G$ that corresponds to conjugation by some
$x\in G$ is called *inner*. An automorphism that isn’t inner is called
an *outer* automorphism.

The composition operation gives the set of all automorphisms of $G$
the structure^{} of a group, $\mathrm{Aut}(G)$. The inner
automorphisms also form a group, $\mathrm{Inn}(G)$, which is a
normal subgroup^{} of $\mathrm{Aut}(G)$. Indeed, if ${\varphi}_{x},x\in G$ is an inner automorphism and $\pi :G\to G$ an arbitrary
automorphism, then

$$\pi \circ {\varphi}_{x}\circ {\pi}^{-1}={\varphi}_{\pi (x)}.$$ |

Let us also note that the mapping

$$x\mapsto {\varphi}_{x},x\in G$$ |

is a surjective^{} group homomorphism with kernel
$\mathrm{Z}(G)$, the centre subgroup^{}. Consequently,
$\mathrm{Inn}(G)$ is naturally isomorphic to the quotient of
$G/\mathrm{Z}(G)$.

Note: the above definitions and assertions hold, mutatis mutandi, if we define the conjugation action of $x\in G$ on $B$ to be the right action

$$y\mapsto {x}^{-1}yx,y\in G,$$ |

rather than the left action given above.

Title | inner automorphism |
---|---|

Canonical name | InnerAutomorphism |

Date of creation | 2013-03-22 12:49:53 |

Last modified on | 2013-03-22 12:49:53 |

Owner | rmilson (146) |

Last modified by | rmilson (146) |

Numerical id | 12 |

Author | rmilson (146) |

Entry type | Definition |

Classification | msc 20A05 |

Synonym | inner |

Defines | conjugation |

Defines | outer |

Defines | outer automorphism |

Defines | automorphism group |