# Group Theory inner automorphism

1. Oct 31, 2013

### Lee33

The problem statement, all variables and given/known data

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?

2. Oct 31, 2013

### Office_Shredder

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

3. Oct 31, 2013

### pasmith

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

4. Oct 31, 2013

### 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?

5. Oct 31, 2013

### Dick

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.

Last edited: Oct 31, 2013