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