Prove that the group of all isometries is abelian

[Jamin2112]

Homework Statement

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.

Homework Equations

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.

 

[Jamin2112]

Hi, guys! Apparently I read the question wrong. The question was "Is the group [Gamma] of all isometries abelian?" It isn't abelian and the proof is quite easy.