Group Theory inner automorphism

[Lee33]

Homework Statement

How do I prove that the inner automorphisms is isomorphic to $S_3$?


The attempt at a solution

I know $S_3 = \{f: \{ 1,2,3 \}\to\{ 1,2,3 \}\mid f\text{ is a permutation}\}$ and I know for every group there is a map whose center is its kernel so the center of of $S_3$ is trivial therefore $S_3/Z(S_3) = 6$.

So an inner automorphism of a group $G$ is an automorphism of the form $ρ_g :x↦gxg^{-1}.$ What I am having trouble with is verifying that distinct elements of $S_3$ give distinct inner automorphism. How can I prove this problem directly?

 

[Office_Shredder]

The inner automorphisms of what are supposed to be isomorphic to S₃?

 

[pasmith]

Homework Statement

How do I prove that the inner automorphisms is isomorphic to $S_3$?


The attempt at a solution

I know $S_3 = \{f: \{ 1,2,3 \}\to\{ 1,2,3 \}\mid f\text{ is a permutation}\}$ and I know for every group there is a map whose center is its kernel so the center of of $S_3$ is trivial therefore $S_3/Z(S_3) = 6$.

So an inner automorphism of a group $G$ is an automorphism of the form $ρ_g :x↦gxg^{-1}.$ What I am having trouble with is verifying that distinct elements of $S_3$ give distinct inner automorphism. How can I prove this problem directly?

Can you prove that if $\rho_g = \rho_h$ then $g^{-1}h \in Z(G)$?

 

[Lee33]

Opps, I forgot to add the inner automorphism of $S_3$ is isomorphic to $S_3.$ Sorry for the confusion!

Pasmith - Can you elaborate please?

 

[Dick]

Opps, I forgot to add the inner automorphism of $S_3$ is isomorphic to $S_3.$ Sorry for the confusion!

Pasmith - Can you elaborate please?

There's no need to elaborate. pasmith pretty much spelled it out. Try to prove the hint he gave you. Once you've done that figure out what the center of $S_3$ is. It should click easily after that.