Question:Inner automorphism automorphism

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This question is about inner automorphism and automorphism | See more questions about inner automorphism| 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 gh is the composite of conjugations by g and h. In other words, if c_g denotes conjugation by g, we have c_{gh} = c_g \circ c_h for all g,h \in G.

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