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Abstract Algebra - automorphism

  1. Sep 25, 2007 #1
    I have two problems I would like to discuss.

    1.For any group G prove that the set of inner automorphisms J(G) is a normal subgroup of the set of automorphisms A(G).

    Let A be an automorphism of G. Let [tex]T_{g}[/tex] be an inner automorphism, i.e.
    [tex]xT_{g}=g^{-1}xg[/tex]

    The problem can be reduced to the question whether the following equality is true:
    [tex]xAT_{g}=xT_{g}A[/tex]

    Then expanding using [tex]xT_{g}=g^{-1}xg[/tex] we have:
    [tex]gxAg^{-1}=gxg^{-1}A[/tex]

    However I am having trouble proving this equality. Attempting to use the definition of normal more directly also did not work, i.e. showing that
    [tex]AT_{g}A^{-1}[/tex] is in J(G).

    2.Let G = {e, a, b, ab} where [tex]a^2=b^2=e[/tex] and ab = ba. Determine the set of automorphisms A(G).
    This problem could be handled easily using brute force but I would like some way to narrow down the possible automorphisms. My attempts to solve the problem so far have resulted in way too many calculations.
     
  2. jcsd
  3. Sep 25, 2007 #2

    matt grime

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    2. Really, way to many calculations? There are only 6 possible permutations of a,b,ab to consider.


    1. If you are still strugglng with this, what inner automorphism do you think you ought to get for AT_gA^{-1}? It is T_h for some h, try taking a reasonable guess as to what h ought to be.
     
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