I wanted to make clear just a quick technical thing. If G is a group, N is a normal subgroup, and [itex] \phi_g \in \text{Inn}(G), \phi_g(h) = g h g^{-1} [/itex] then [itex] \phi_g [/itex] is an automorphism of N, right? However, is it the case that we cannot say that [itex] \phi_g [/itex] is an inner automorphism, since we are not guaranteed that [itex] g \in N [/itex]? I think this is the case, but I just want to be clear.(adsbygoogle = window.adsbygoogle || []).push({});

Furthermore, is it then possible to extend any group to a larger group such that all automorphisms are given by inner automorphisms from the larger group? More precisely, if G is a group, is there always an injective homomorphism [itex] G \to H [/itex] such that [itex] \text{Aut}(G) \cong \text{Inn}(H) [/itex] ?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Extending automorphism groups to inner automorphism groups.

Loading...

Similar Threads for Extending automorphism groups |
---|

I Spin group SU(2) and SO(3) |

I What is difference between transformations and automorphisms |

A Extending a linear representation by an anti-linear operator |

I Groups of Automorphisms - Aut(C) ... |

I Automorphisms of Field Extensions ... Lovett, Example 11.1.8 |

**Physics Forums | Science Articles, Homework Help, Discussion**