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Math Help - Group homomorphism

  1. #1
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    Group homomorphism

    Describe a group homomorphism from U_5 to S_4.

    U_5 is the group of units Z/5Z under multiplication
    S_4 is the set of permutations on 4 elements.

    The hint for the exercise said to use Cayley's Theorem.
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  2. #2
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    Quote Originally Posted by Zennie View Post
    Describe a group homomorphism from U_5 to S_4.

    U_5 is the group of units Z/5Z under multiplication
    S_4 is the set of permutations on 4 elements.

    The hint for the exercise said to use Cayley's Theorem.
    I think a group homomorphism is a mapping from group U_5 [with say operation *] into S_4 [with operation *'].

    And f(a*b)=f(a)*'f(b) \forall a,b \in S_4.
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  3. #3
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by Zennie View Post
    Describe a group homomorphism from U_5 to S_4.

    U_5 is the group of units Z/5Z under multiplication
    S_4 is the set of permutations on 4 elements.

    The hint for the exercise said to use Cayley's Theorem.
    Well, what is Cayley's theorem?

    The hint is telling you, basically, to work through the proof of Cayley's theorem using an example. There are a number of different homomorphisms from U_5 to S_4, but the point of this question is not to conjure up these, but to find a specific one using this theorem.

    So, do you have any problems understanding the theorem? It basically says you can find a `copy' of U_5 in S_4.
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