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 , then there is a homomorphism :G given by (g):x gx = (g,x). Conversely, every homomorphism :G ] defines an action, namely, gx = (g)x, which makes X into a G-set.

I am having trouble understanding the formalism of the statement:

"there is a homomorphism :G given by (g):x gx = (g,x). "

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

Peter