Let G be a finite group acting on the set S. Suppose so that for any there is a unique so that . For each , let .
Prove:
I think about trying to use the order since if but other than that I don't know what I need to do.
You made the correct observation. If we can prove that then and so . Now, suppose for a second that then you know but you know that there is a UNIQUE which does this. What could this be?
You made the correct observation. If we can prove that then and so . Now, suppose for a second that then you know but you know that there is a UNIQUE which does this. What could this be?
You have which is true if but the unique h is for where aren't necessarily the same.