Inner automorphisms of $S_3$ 
How do I prove that $S_3 \simeq \wp(S_3)$? So I must show that the group of inner automorphisms of $S_3$ is isomorphic to $S_3$. 

I haven't been given many examples on how to do these types of questions but I know if someone can provide an explicit answer I will know how to do similar questions.
 A: My goal is to prove that $S_3$ is isomorphic to the group of inner automorphisms of $S_3$. Well, we can define a map $\phi$: $S_3$ ---> $\gamma$, where $\phi$(a) ---> $\gamma$(a). Define $\gamma$ as an inner automorphism. $\gamma$: a--->$gag^{-1}$. Now, since Z($S_3$), the center of $S_3$, is trivial, we have that the kernel of $\phi$ is (1). Now, this implies that $\phi$ is injective, so, it must map to at least 6 elements. Also note that $\phi$ is a homomorphism: we have : $\phi$(a)$\phi$(b)= $gag^{-1}$$gbg^{-1}$ = $gabg^{-1}$ = $\phi$(ab). $\phi$ is surjective: let $\gamma$: a--->$gag^{-1}$ be some arbitrary inner automorphism of $S_3$. Then, $\gamma$ = $\phi$(a). Therefore,$\phi$ is an isomorphism between $S_3$ and the group of inner automorphisms of $S_3$
Side note: Since $\phi$ is a homomorphism it must preserve the order of elements. Therefore, every element of order 2, (of which there are three), must be mapped to an element of order 2. Note that this implies there are 6 possible mappings for each transposition in $S_3$.
A: Hint: You have a map $S_3\to S_6$ which sends $x$ to its associated inner automorphism. The image of this map is obviously surjective onto the inner automorphisms. What is its kernel?
