# Thread: Prove Inn(G) is isomorphic to G

1. ## Prove Inn(G) is isomorphic to G

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)

2. Originally Posted by page929
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)

You are correct...so far: you still haven't proved thaty $Inn(G)\cong G$ ...

Tonio

3. Tonio -

Can you help me finish? Where do I go from here?

Thanks

4. any surjective homomorphism on a finite set is injective because....

5. Originally Posted by Deveno
any surjective homomorphism on a finite set is injective because....

Any surjective map from a finite set to itself (or to a set which has the same number of elements) is injective,

but here we still don't know whether $|Inn(G)|=|G|$ ...

Tonio

6. Originally Posted by page929
Tonio -

Can you help me finish? Where do I go from here?

Thanks

Hints:

1) First prove in general that for any group $G\,,\,\,G/Z(G)\cong Inn(G)$ , using the same map you did in

2) Now prove directly that $Z(S_3)=\{1\}$

Tonio

7. 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.

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# prove that g/z(g) is isomorphic to inn(g)

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