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Math Help - Homomorphisms, Integers mod 6 to Symmetry gp

  1. #1
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    Homomorphisms, Integers mod 6 to Symmetry gp

    There is a unique homomorphism \theta: \mathbb{Z}_{6}\rightarrow S_{3} such that \theta ([1]) =(1 \ 2 \ 3). Determine \theta ([k]) for each [k] \in \mathbb{Z}_{6}. Which elements are in Ker \theta ?

    I don't understand how that single condition can be used to determine the rest of the \theta ([k]). It doesn't seem like enough information.
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    Quote Originally Posted by Blue Calx View Post
    There is a unique homomorphism \theta: \mathbb{Z}_{6}\rightarrow S_{3} such that \theta ([1]) =(1 \ 2 \ 3). Determine \theta ([k]) for each [k] \in \mathbb{Z}_{6}. Which elements are in Ker \theta ?

    I don't understand how that single condition can be used to determine the rest of the \theta ([k]). It doesn't seem like enough information.
    If \theta (1) = (123) then \theta (2) =\theta(1+1) = (123)^2.
    And \theta(3) = \theta(1+1+1) = (123)^3.
    And so on ...
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