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.
yes, i know that he has to show that |Inn(G)| = |G|. and it's probably easier to show that S3/Z(S3) is isomorphic to S3, than to prove ia = ib --> a= b.
alternatively, he could show that the only inner automorphism equal to 1G, the identity map on G, is ie.