By analyzing the map $G \rightarrow$ Aut(G) given by $g \rightarrow \phi_g$, identify Inn(G) as a quotient of G.

Question

By analyzing the map $G \rightarrow\operatorname{Aut}(G)$ given by $g \rightarrow \phi_g$, identify $\operatorname{Inn}(G)$ as a quotient of $G$.

($\phi_g\colon G \longrightarrow G$ by $\phi(x) = gxg^{-1}$, $\operatorname{Inn}(G)$ being the set of inner automorphisms)

I'm not entirely sure what they mean by identify $\operatorname{Inn}(G)$ as a quotient of $G$”. Do they mean find a subgroup $K$ such that $G/K = \operatorname{Inn}(G)$? It seems very vague to me.

Answer 1

It is not vague. The image of your map (let us call it $\phi$) is, by definition, $\operatorname{Inn}G$. Therefore, $\operatorname{Inn}G\simeq G/\ker\phi$. Note that $\ker\phi$ is the center of $G$.