Hello, It will be easier to first post the question: [Note: there are many parts to the original question. This is only part of it]. I am not sure how to proceed here. Firstly I know that the above function is a translation. I also know that if delta has an infinite order then deltan cannot equal the identity for any n which is an element of Z. Now I can see that x deltan = (x+n, (-y)n) and that deltan will never return to (x,y). But I am not sure how to prove this. If anything is unclear, please ask and I will try to clarify my question. Do I first prove by induction on n that x deltan = x + n for all x, n > 0 and if n > 0 then x + n is not equal to x so deltan does not equal the identity? Any help is appreciated. Thankyou.