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 $\alpha$, then there is a homomorphism $\widetilde{\alpha}$:G $\longrightarrow$ $S_X$ given by $\widetilde{\alpha}$(g):x $\longrightarrow$gx = $\alpha$(g,x). Conversely, every homomorphism $\phi$:G $\longrightarrow$] $S_X$defines 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 $\longrightarrow$gx = $\alpha$(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 $\longrightarrow$gx = $\alpha$(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

2. ## 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").