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Homework Help: Group theory

  1. May 30, 2007 #1
    1. The problem statement, all variables and given/known data

    Prove that [tex]Aut(S_3)=S_3[/tex]
    2. Relevant equations
    = means isomorphic

    3. 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?

  2. jcsd
  3. May 30, 2007 #2


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    Homework Helper

    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:

  4. May 31, 2007 #3

    matt grime

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    Science Advisor
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    No. This does not prove anything.
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