Prove that the group of all isometries is abelian

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In summary, the conversation discusses proving that isometric functions commute and the properties that prove the set of isometric functions is a group. The homework equation states that a bijective function that preserves distances is isometric. The attempt at a solution involves showing that for two isometric functions and an integer, their compositions also commute. The conversation ends with a hint to consider whether the group of all isometries is abelian.
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Jamin2112
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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.
 
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  • #2
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.
 

What is the definition of an isometry?

An isometry is a transformation that preserves the distance and angles between points in a geometric figure.

What is a group in mathematics?

In mathematics, a group is a set of elements with a binary operation that satisfies the four group axioms: closure, associativity, identity, and inverse.

Why is the group of all isometries important?

The group of all isometries is important because it represents all the possible symmetries of a geometric figure. It is also used in various fields of mathematics, such as geometry, topology, and physics.

Why is it important to prove that the group of all isometries is abelian?

Proving that the group of all isometries is abelian is important because it helps us understand the structure and properties of this group. It also allows us to use theorems and techniques from abelian groups to study the group of all isometries.

How do you prove that the group of all isometries is abelian?

To prove that the group of all isometries is abelian, we need to show that for any two isometries A and B, their composition AB is equal to BA. This can be done by using the definition of isometries and the properties of geometric figures.

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