Prove that the group of all isometries is abelian

  • Thread starter Jamin2112
  • Start date
  • #1
986
9

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.
 

Answers and Replies

  • #2
986
9
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.
 

Related Threads on Prove that the group of all isometries is abelian

  • Last Post
Replies
9
Views
4K
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
14
Views
2K
  • Last Post
Replies
9
Views
6K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
0
Views
2K
Replies
1
Views
935
Replies
5
Views
896
  • Last Post
Replies
1
Views
833
Top