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Math Help - Isomorphic groups

  1. #1
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    Isomorphic groups

    Prove that Aut(V) is isomorphic to S_3 and that Aut(S_3) is isomorphic to S_3. V is the four group.

    Also, prove that Aut(Z) is isomorphic to I_2. Z is the set of integers and I_2 is the integers mod 2.

    I appreciate any help.
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  2. #2
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    1) Any automorphism on V will preserve order of each elements. Therefore it can permute i,j,k. There are no restrictions on the permutation. Hence Aut(V)= S_3.

    2) Any isomorphism from the integers will have the form \theta (a+b) = \theta (a)+\theta (b) (*)

    \theta (0) = 0

    Let \theta(1) = c

    Then for all positive integers \theta (n) = nc (induction using (*)

    And for negative integers \theta (-n) = -nc

    For this to be a bijection c = \pm 1 and we're done
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  3. #3
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    Quote Originally Posted by didact273 View Post
    Prove that Aut(V) is isomorphic to S_3 and that Aut(S_3) is isomorphic to S_3. V is the four group.

    Also, prove that Aut(Z) is isomorphic to I_2. Z is the set of integers and I_2 is the integers mod 2.

    I appreciate any help.
    Remember that S_3 = \{ e,a,a^2,b,ab,a^2b\} where a=(123),b=(12). Any automorphism on S_3 is determined by \theta(a),\theta(b). The order of \theta(a) is equal to order of a, thus, \theta(a) = a,a^2 similarly \theta(b) = b,ab,a^2b. In total we have at most 6 automorphism. Show that each one extens to an automorphism of S_3.
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  4. #4
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    Quote Originally Posted by ThePerfectHacker View Post
    Remember that S_3 = \{ e,a,a^2,b,ab,a^2b\} where a=(123),b=(12). Any automorphism on S_3 is determined by \theta(a),\theta(b). The order of \theta(a) is equal to order of a, thus, \theta(a) = a,a^2 similarly \theta(b) = b,ab,a^2b. In total we have at most 6 automorphism. Show that each one extens to an automorphism of S_3.
    I understand everything until the last part. Can you explain what you mean by each automorphism extending to an automorphism in S_3? An example of one of them?

    Thanks.
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  5. #5
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    Quote Originally Posted by didact273 View Post
    I understand everything until the last part. Can you explain what you mean by each automorphism extending to an automorphism in S_3? An example of one of them?

    Thanks.
    I mean if \theta(a) = a^2,\theta(b)=ab. Then \theta(a^2) = (a^2)^2=a and \theta(ab) = \theta(a)\theta(b) = (a^2)(ab) = b. So on and so forth. You will need to show that when you extend \theta, then \theta would be an automorphism.
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