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.
If f:Z-->Z is bijective and preserves distances, then f is isometric.
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.