Proving Aut(S_3) is Isomorphic to S_3

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In summary, the conversation discusses how to prove that Aut(S_3), the set of isomorphisms of S_3 onto itself, is equal to S_3. It is suggested that showing there are exactly 6 isomorphisms would prove this, but this cannot be proven simply by noting that they have the same order. The conversation then explores using the relation (12)(13)=(132) to find a pair of isomorphisms that don't commute, which would differentiate between the groups of order 6: Z_6 and S_3.
  • #1
happyg1
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Homework Statement




Prove that [tex]Aut(S_3)=S_3[/tex]

Homework Equations


= means isomorphic


The Attempt at a Solution



If I let [tex]S_3[/tex] 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
 
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  • #2
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)
 
  • #3
happyg1 said:
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.
 

FAQ: Proving Aut(S_3) is Isomorphic to S_3

How do you prove Aut(S3) is isomorphic to S3?

In order to prove that two groups are isomorphic, we need to show that there exists a bijective homomorphism between them. In this case, we need to find a bijective homomorphism between Aut(S3) and S3. This can be done by showing that the two groups have the same order and the same group structure.

What is the order of Aut(S3) and S3?

The order of a group is the number of elements in that group. In this case, the order of Aut(S3) is 6, as there are 6 automorphisms of S3. The order of S3 is also 6, since there are 6 elements in S3.

What is an automorphism?

An automorphism is a bijective homomorphism from a group to itself. In other words, it is a function that maps a group onto itself and preserves the group structure. In the context of Aut(S3), an automorphism is a function that maps S3 onto itself and preserves the group structure of S3.

What is the group structure of Aut(S3) and S3?

The group structure of a group refers to the way in which the elements of the group combine together under the group operation. In this case, the group structure of Aut(S3) is the same as that of S3. Both groups have the same group operation, which is composition of functions.

Why is it important to prove that Aut(S3) is isomorphic to S3?

Proving that Aut(S3) is isomorphic to S3 is important because it allows us to understand the properties and structure of both groups better. It also helps us to generalize our understanding of automorphisms and isomorphisms to other groups. Additionally, this proof can be applied to other groups, leading to a deeper understanding of group theory.

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