Question:Inner automorphism 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 $$\operatorname{Inn}(G) \cong G/Z(G)$$.