Show that for G = S3, Inn(G) G.

Here is what I have:

Construct φ: G à Inn(G) by φ(a) = ia a G.

(Prove φ is a homomorphism)

Pick a G

Then φ(ab) = iab

But iab(x) = abx(ab)-1

= abxb-1a-1

= a(bxb-1)a-1

= ia(bxb-1)

= ia(ib(x))

So, iab = iaib

Therefore, φ(ab) = φ(a)φ(b)

And φ is surjective by the definition of Inn(G)

Can anyone let me know if I am correct? If not, please help.