View source for Question:Inner automorphism automorphism

{{quotation|This question is about [[question about::inner automorphism]] and [[question about::automorphism]] <nowiki>|</nowiki> {{#ask: [[question about::inner automorphism]]|limit = 0|searchlabel = See more questions about inner automorphism}}<nowiki>|</nowiki> {{#ask: [[question about::automorphism]]|limit = 0|searchlabel = See more questions about automorphism}}}}

Q: '''Why is an [[inner automorphism]] an [[automorphism]]? How is the group structure of a group related to that of its [[automorphism group]]?'''

A: This basically is a straightforward verification that involves some symbol manipulation. See [[group acts as automorphisms by conjugation]], which not only shows that conjugations by elements are automorphisms, but also that the conjugation by <math>gh</math> is the composite of conjugations by <math>g</math> and <math>h</math>. In other words, if <math>c_g</math> denotes conjugation by <math>g</math>, we have <math>c_{gh} = c_g \circ c_h</math> for all <math>g,h \in G</math>.

We thus obtain a [[homomorphism of groups]] from <math>G</math> to <math>\operatorname{Aut}(G)</math> (the [[automorphism group]] of <math>G</math>) sending each <math>g \in G</math> to the conjugation map <math>c_g</math>. The kernel of this is the [[center]] <math>Z(G)</math> and the image is the [[inner automorphism group]] <math>\operatorname{Inn}(G)</math>. We thus have, from the [[first isomorphism theorem]], that <math>\operatorname{Inn}(G) \cong G/Z(G)</math>.