1. The problem statement, all variables and given/known data The only thing I need to do now is show that isometric functions commute. I've shown the 3 properties that prove the the set G of isometric functions is a group. 2. Relevant equations If f:Z-->Z is bijective and preserves distances, then f is isometric. 3. The attempt at a solution f, g in G and n in Z --------> |f(n) - g(n)| = |f(f(n)) - g(f(n))| = |g(f(n) - g(g(n))| --------> ????????? --------> |f(g(n) - g(f(n))| = 0 --------> f(g(n)) = g(f(n)). QED. Can you give me an oh-so-subtle hint? I just started class this week, so I'm not yet in my proper mindset.