What are the automorphism groups of S3?

[hgj]

The question is to determine the group of automorphisms of S3 (the symmetric group of 3! elements).

I know Aut(S3)=Inn(S3) where Inn(S3) is the inner group of the automorphism group.

For a group G, Inn(G) is a conjugation group (I don't fully understand the definition from class and the book doesn't give one).

I also know that Inn(S3) is a subgroup of Aut(S3), so Inn(S3) is contained in
Aut(S3).

I'm not sure how to show the other way thought, that everything in Aut(S3) is in Inn(S3).

I talked to my professor, and he gave me the hint that Inn(S3) has only 6 automorphisms, so I should show there are no more than 6 automorphisms in
Aut(G). The problem is I'm not sure how to do this.

 

[BerkMath]

A basic result on the subject, which I am sure would be found in any book or article pertaining to such, is that for n not equal to 6, Inn(Sn) exhausts the Aut(Sn) i.e. they are the same. I think its called complete (one of the many uses of the word). In other words for n not equal to 6, Out(Sn) is trivial. I recall briefly the proof. The idea is to show that all automorphisms that preserve the conjugacy class of transpositions is inner. Then you show for that for all n not 6, there does not exsist a conjugacy class of order 2 with the same order as the class of tranpositions. After this the proof is trivial since Inn(S3) is comprised of the mappings which conjugate all elements of S3 by a fixed element in S3. There are only 3!=6 such elements for which to fix, and therefore the order of Inn(S3)=6.

 

[matt grime]

S_3 is the permutation group on 3 elements, it has order 3!.consider where an automorphism sends the 2-cycles (there are 3; what groups of order 3! do you know?)

there is no need to invoke the aut/inn general S_n argument for this case, it can be done by inspection.

oh, and inner automorphism of G is a map f_x from G to G f_x(g)=xgx^{-1}

this is an automorphism, and f_x=f_y means that x=y, thus there are distinct automorphisms for each element of G

 

[AKG]

Hints:

Find a set of two generators for S₃. Actually, find all pairs of generators for the group. What do all these pairs have in common? You should find two types of pairs. Specifically, you will find 9 pairs of generators, 3 will be of one type, 6 will be of another. Pick one type.

Inn(S₃) = Aut(S₃).

Determine the relationship between whether or not two permutations have the same cycle structure and whether or not they are conjugate.

Having picked your type in the first hint, if {x,y} is a pair of elements, and {x',y'} is any pair of the same type, is x -> x', y -> y' an automorphism?