Chapter 4.6, Problem 32E

Let $a$ be a fixed element of the group $G$. According to Exercise 20 of section 3.5, the mapping $t_a:G\to G$ defined by $t_a(x)=ax{a}^{-1}$ is an automorphism of $G$. Each of these automorphism $t_a$ is called an inner automorphism of $G$. Prove that the set $Inn(G)=\left\{t_a\right|a\in G\}$ forms a normal subgroup of the group of all automorphism of $G$.

Exercise 20 of Section 3.5

For each $a$ in the group $G$, define a mapping $t_a:G\to G$ by $t_a(x)=ax{a}^{-1}$. Prove that $t_a$ is an automorphism of $G$.