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Math Help - Automorphism proof

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    Automorphism proof

    Prove that  Aut (Z_{2}XZ_{2}) \cong S_{3} .
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    Quote Originally Posted by pirouette View Post
    Prove that  Aut (Z_{2}XZ_{2}) \cong S_{3} .
    Let \phi: \mathbb{Z}_2\times \mathbb{Z}_2\to \mathbb{Z}_2\times \mathbb{Z}_2 be an automorphism.
    Obviously, \phi(0,0) = (0,0).

    1)Now, \phi(0,1) has to be an element of order 2 and so \phi(0,1) = (0,1) \text{ or }(1,0)\text{ or }(1,1).

    2)The same possibilities for \phi(0,1) however except for what was used in #1.

    3)Once #1 and #2 are determined then \phi(1,1) = \phi(1,0) + \phi(0,1) and so \phi(1,1) is determined.

    There are 3 possibilities for #2 and 2 possibilities #3, we therefore have at most 2\cdot 3 = 6 automorphisms.

    Check that all these six instead give a raise to a hextic automorphism group.
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