Let $G$ be a group of order $n$. Define what is meant by a permutation representation of $G$. Using such representations, show $G$ is isomorphic to a subgroup of the symmetric group $S_{n}$. Assuming $G$ is non-abelian simple, show $G$ is isomorphic to a subgroup of $A_{n}$. Give an example of a permutation representation of $S_{3}$ whose kernel is $A_{3}$.