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Math Help - Actions and G-Sets

  1. #1
    Super Member Bernhard's Avatar
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    Actions and G-Sets

    I am a math hobbyist working alone. I am reading "An Introduction to the Theory of Groups" by Joseph Rotman.

    Theorem 3.18 in the section on G-Sets reads as follows:

    If X is a G-set with action \alpha, then there is a homomorphism \widetilde{\alpha}:G \longrightarrow S_X given by \widetilde{\alpha}(g):x \longrightarrowgx = \alpha(g,x). Conversely, every homomorphism \phi:G \longrightarrow] S_Xdefines an action, namely, gx = \phi(g)x, which makes X into a G-set.

    I am having trouble understanding the formalism of the statement:

    "there is a homomorphism \widetilde{\alpha}:G \longrightarrow S_X given by \widetilde{\alpha}(g):x \longrightarrowgx = \alpha(g,x). "

    \widetilde{\alpha}:G \longrightarrow S_X gives \widetilde{\alpha} as a mapping from G to S_X which would be of the form g \longrightarrow \sigma_g where g \in G and \sigma_g \in S_X whereas \widetilde{\alpha}(g):x \longrightarrowgx = \alpha(g,x) is referring to a mapping from X to X. So it seems \widetilde{\alpha} is being explained in terms of its effect on X while being a mapping from G to S_X. This seems confusing. Can anyone please clarify?

    Peter
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  2. #2
    MHF Contributor

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    Re: Actions and G-Sets

    it's just a notational issue. the image of \widetilde{\alpha} is an element of S_X, and the elements of S_X are themselves mappings from X to X.

    so one ought to write: \widetilde{\alpha}(g) = the mapping that takes x \to \alpha(g,x).

    so instead of the image of \widetilde{\alpha} being a "point" it's an "arrow".

    to amplify, usually the action \alpha is "hidden", one just defines what you mean by g.x (often written as simply gx, or sometimes x → g(x), which is what happens when you call \sigma_g "the function g").
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