Chapter 4.2, Problem 11E

For each $a$ in the group $G$, define a mapping $k_a:G\to G$ by $k_a(x)=x{a}^{-1}$ for all $x$ in $G$.

a. Prove that each $k_a$ is a permutation on the set of elements in $G$.

b. Prove that $H=\left\{k_a|a\in G\right\}$ is a group with respect to mapping composition.

c. Define $\varphi :G\to K$ by $\varphi (a)=k_a$ for each $a$ in $G$.Determine whether $\varphi$ an isomorphism is always. This mapping $\varphi$ is known as the right regular representation of $G$.