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Math Help - Left Representation

  1. #1
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    Left Representation

    Let G be a finite group and let g \in G. Let m be the order of g. Prove that in the representation G \to S_{|G|} (given by left multiplication) every cycle in the cycle decomposition of the image of g has length m.
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    MHF Contributor Drexel28's Avatar
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    Re: Left Representation

    Quote Originally Posted by kierkegaard View Post
    Let G be a finite group and let g \in G. Let m be the order of g. Prove that in the representation G \to S_{|G|} (given by left multiplication) every cycle in the cycle decomposition of the image of g has length m.
    The basic idea is that if one selects x\in G then one has the m-cycle (x,gx,\cdots,g^{m-1}x), then selecting y\in G-\{x,gx,\cdots,g^{m-1}x) one has the m-cycle (y,gy,\cdots,g^{m-1}y). I think you get the idea now.
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    Re: Left Representation

    let H = <g>, and \sigma_g be the image of g in the representation. what is the action of \sigma_g on the right cosets Hx?

    how is this relevant?
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