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 , 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?
Re: Actions and G-Sets
it's just a notational issue. the image of is an element of , and the elements of are themselves mappings from X to X.
so one ought to write: the mapping that takes .
so instead of the image of being a "point" it's an "arrow".
to amplify, usually the action 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 "the function g").