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Math Help - Automorphisms

  1. #1
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    Automorphisms

    Find Aut(Z).

    I understand kind of what an automorphism is (and isomorphism from a group G onto itself), and I understand what an isomorphism is (a one-to-one mapping from G to G-bar that preserves the group operation), but we were never given any examples of finding an automorphism. I feel like this shouldn't be very hard, but I can't figure it out!

    Please help!!
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  2. #2
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    Quote Originally Posted by bluejay View Post
    Find Aut(Z).

    I understand kind of what an automorphism is (and isomorphism from a group G onto itself), and I understand what an isomorphism is (a one-to-one mapping from G to G-bar that preserves the group operation), but we were never given any examples of finding an automorphism. I feel like this shouldn't be very hard, but I can't figure it out!

    Please help!!
    Is 'Z' referring to the set of all integers? If not, what is the set 'Z'?
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  3. #3
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    Yes, Z is referring to the set of all integers.

    I think seeing an example of finding an automorphism of any group would be helpful...
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  4. #4
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    If \phi : \mathbb{Z}\to \mathbb{Z} is an automorphism it is completely determined by knowing \phi (1) since 1 generates the group. Let \phi(1) = n where n\in \mathbb{Z}. Then \phi ( x ) = nx. It can be shown for \phi to be one-to-one and onto we require n=\pm 1. Thus, \phi_1 (x) = x and \phi_2 (x) = -x are the only two automorphisms.
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