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

    I am having trouble understanding the formalism of the statement:

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

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

    Peter
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  2. #2
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    Re: Actions and G-Sets

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

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

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

    to amplify, usually the action $\displaystyle \alpha$ 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 $\displaystyle \sigma_g$ "the function g").
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