Chapter 3.5, Problem 20E

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 $G$.

Sec. $4.6,\#32\ll$

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 automorphisms $t_a$ 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 automorphisms of $G$.