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Math Help - automophism proof

  1. #1
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    automophism proof

    Let H and K be groups with relatively prime orders. Show that Aut(HxK)
    Aut(H)xAut(K).
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  2. #2
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    Re: automophism proof

    Quote Originally Posted by henderson7878 View Post
    Let H and K be groups with relatively prime orders. Show that Aut(HxK)
    Aut(H)xAut(K).
    Given \varphi \in \text{Aut}(H) and \psi \in \text{Aut}(K), define a map \text{Aut}(H) \times \text{Aut}(K) \to \text{Aut}(H\times K) by f: (\varphi,\psi)(h,k)\mapsto (\varphi(h),\varphi(k)). Show that this is a well defined homomorphism first, then show that it is bijective.
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  3. #3
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    Re: automophism proof

    Hi,
    This really isn't very hard, but it is kind of messy. The previous answer is a little misleading in that it looks as though Aut(H) cross Aut(K) would always be isomorphic to Aut(H cross K), which of course is false. Here's the mess:

    automophism proof-mhfautomorphisms.png

    automophism proof-mhfautomorphisms_a.png
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