1. ## 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$$\displaystyle S_X given by \displaystyle \widetilde{\alpha}(g):x\displaystyle \longrightarrowgx = \displaystyle \alpha(g,x). Conversely, every homomorphism \displaystyle \phi:G\displaystyle \longrightarrow]\displaystyle S_Xdefines an action, namely, gx = \displaystyle \phi(g)x, which makes X into a G-set. I am having trouble understanding the formalism of the statement: "there is a homomorphism \displaystyle \widetilde{\alpha}:G\displaystyle \longrightarrow$$\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

2. ## 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").