Assume G is a group and X is a set. and mu: G times X rightarrow X is an action.

Question

Assume G is a group and X is a set. and mu: G times X rightarrow X is an action. Let R = {f: X rightarrow R} be the set of all functions from X to R. Define m: G times R rightarrow R be the function m (g, f) = f compositefunction mu_g - 1. a) Prove that m is an action of G on R. b) (Bonus) Prove that there is a bisection between R^G and {f: x/G rightarrow R}, the set of functions from X/G to R. (This is a primitive version of Noether's theorem, i.e., for every group action of a physical system, there is a corresponding conservation law.)

Explanation / Answer

in this problem statement

bijection function from f to g should be defined

also invertibility criteria of fog should be clearly stated here in this problem

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