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Math Help - construct an epimorphism

  1. #1
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    construct an epimorphism

    Hi,
    i'm trying to construct an construct an epimorphism from S_3 \times \mathbb{Z} \rightarrow \mathbb{Z} and from S_3 \rightarrow C_2 but unfortunately i have no idea how to start...
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  2. #2
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    Re: construct an epimorphism

    For a in S_3 and n in Z, map (a, n) to n. That's trivial.

    For S_3\to C_2 what do you mean by " S_3" and " C_2"? My first thought was that " S_3" was the set of permutations on 3 objects and " C_2" was the set of pairs of complex numbers but in that case, S_3 is finite (containing 6 members) while C_2 is infinite so there cannot be such an epimorphism.
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  3. #3
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    Re: construct an epimorphism

    You haven't given enough information. The morphism (\sigma,n) \mapsto n is an epimorphism for the first one. The morphism \sigma \mapsto \text{sign}(\sigma) is an epimorphism for the second one. I am not sure what more you are trying to do with this. If you want, you can use (\sigma,n) \mapsto \text{sign}(\sigma)\cdot n which would also be an epimorphism for the first one.
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  4. #4
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    Re: construct an epimorphism

    Quote Originally Posted by HallsofIvy View Post
    For a in S_3 and n in Z, map (a, n) to n. That's trivial.

    For S_3\to C_2 what do you mean by " S_3" and " C_2"? My first thought was that " S_3" was the set of permutations on 3 objects and " C_2" was the set of pairs of complex numbers but in that case, S_3 is finite (containing 6 members) while C_2 is infinite so there cannot be such an epimorphism.
    Typically, Z_n is the cyclic group of order n, but I imagine that is the group the OP meant.
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  5. #5
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    Re: construct an epimorphism

    Quote Originally Posted by HallsofIvy View Post
    For a in S_3 and n in Z, map (a, n) to n. That's trivial.

    For S_3\to C_2 what do you mean by " S_3" and " C_2"? My first thought was that " S_3" was the set of permutations on 3 objects and " C_2" was the set of pairs of complex numbers but in that case, S_3 is finite (containing 6 members) while C_2 is infinite so there cannot be such an epimorphism.

    maybe this?

    \text{Let }C_\infty\text{ be the group }(Z,+)\text{, and, for an integer }m\geq1\text{, let }C_m\text{ be the group }(Z/mZ,+).

    from here

    That's Integers Z, I didn't notice a Tex code for it.
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  6. #6
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    Re: construct an epimorphism

    An explicit epimorphism \phi: S_3 \to C_2, where C_2 = \{e,a\}:

    \phi(e) = e

    \phi((1\ 2)) = a

    \phi((1\ 3)) = a

    \phi((2\ 3)) = a

    \phi((1\ 2\ 3)) = e

    \phi((1\ 3\ 2)) = e

    Proving this is a homomorphism is the tricky part, it may be easier to observe that what we are actually doing is mapping:

    S_3 \to S_3/A_3, which is isomorphic to any cyclic group of order 2, since it is of order 2 (any two groups of prime order are isomorphic and both are cyclic).
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