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Math Help - Prove this mapping is a homomorphism

  1. #1
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    Prove this mapping is a homomorphism

    Here is the example:
    The mapping from S_n to Z_2 that takes an even permutation to 0 and an odd permutation to 1 is a homomorphism.

    -I have to prove that this example is a homomorphism.

    I have looked at other homomorphisms and I understand how to prove them. They are pretty easy, such as determinants or derivatives. For some reason this one just eludes me, and I feel that there is a simple answer. i know that i have to prove phi(ab)=phi(a)phi(b) . this is an additive group so i think it would be phi(a+b)=phi(a)+phi(b) right? and i maybe have to do two cases one for even, one for odd? Thanks for help in advance.
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  2. #2
    MHF Contributor

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    Tejas
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    Re: Prove this mapping is a homomorphism

    no, the group operation in S_n is composition, so what you want to prove is:

    \varphi(\sigma \circ \tau) = \varphi(\sigma) + \varphi(\tau), for \sigma,\tau \in S_n.

    what this boils down to is proving:

    even composed with even is even
    even composed with odd is odd
    odd composed with even is odd
    odd composed with odd is even
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