Proving Aut(S_3) is Isomorphic to S_3

[happyg1]

Homework Statement

Prove that $$Aut(S_3)=S_3$$

Homework Equations

= means isomorphic

The Attempt at a Solution

If I let $$S_3$$ be {1,2,3} then I can write out explicitly its 6 elements...the permutations of 1,2,3...
Aut(S3) is the set of isomorphisms of S3 onto itself. So can I just write them all out and then say that since they have the same order they are isomorphic?
Or is there a better way?

Thanks,
CC

 

[StatusX]

If you can show there are exactly 6 isomorphisms, then you've shown Aut(S_3) is one of the two groups of order 6: Z_6 and S_3. These can be distinguished by the fact that Z_6 is abelian while S_3 is not, so it only remains to find a pair of isomorphisms that don't commute.

How were you planning on showing there are exactly 6 isomorphisms? If you're not sure here, think about the relation:

(12)(13)=(132)

 

[matt grime]

Aut(S3) is the set of isomorphisms of S3 onto itself. So can I just write them all out and then say that since they have the same order they are isomorphic?

No. This does not prove anything.