Every finite subgroup of isometries of n-dimensional space

In summary, a "rigid motion" or isometry is a function that preserves distances between points. A symmetry is a type of rigid motion that leaves an object unchanged. The group of symmetries on a finite set can either be the dihedral group or a cyclic group. The Orthogonal group, O(n), fixes the origin, meaning every isometry in O(n) leaves the origin fixed. The elements of O(n) are maps/matrices. In a vector space, an isometry is also an affine map. This is used in the proof that the group of isometries on a finite set is either the dihedral group or a cyclic group.
  • #1
Battousii
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0
... fixes at least one point.

I recently came upon the proof in a book and I didn't quite understand the notion of "rigid motion", and I was wondering if you could help clarify it for me. Is it just "the vertices must stay in the given order", as used in symmetries of polygons? I've attached the proof.

Thanks.
 

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  • #2
A "rigid motion" is also called an isometry. An isometry is a function [tex]f:\mathbb{R}^n \mapsto \mathbb{R}^n[/tex] so that [tex]|\bold{x}-\bold{y}| = |f(\bold{x}) - f(\bold{y})|[/tex]. Where [tex] | \ |[/tex] is the Euclidean metric. All this means it is a function such that the distance between any two points remains the same after the function is performed. Thus, in a sense it is "rigid", i.e. it leaves the object in tact. So reflections, rotations, translations are simple examples of functions which leave objects unchanged.

Given a set (or an object) in space a "symettry" is a rigid motion which leaves the object unchanged, i.e. it is its own image under the map. In that case we can define a "symettery group" on an object as the set of all symettries on the object.

I think what you are trying to prove is that the group of symettries on a finite set is either the dihedral group or a cyclic group.
 
  • #3
Alright, thank you. That makes sense. Basically, a rigid motion will preserve distances and angles (in terms of vectors).

Can you clarify what it means when the Orthogonal group, [itex]O(n)[/itex], fixes the origin?
 
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  • #4
It means every isometry in O(n) leaves 0 fixed. The elements of O(n) aren't 'vectors' - they're maps/matrices.
 
  • #5
What they are saying in the book is this:

Let G be your finite group of isometries/rigid motions on a vector space V. Let v ∈ V be arbitrary. Define

[tex]w := \frac{1}{G} \sum_{g \in G} g(v)[/tex]

now for any h ∈ G, we have h(w)=w:

[tex]h(w) = h\left( \frac{1}{G} \sum_{g \in G} g(v) \right) = \frac{1}{G} \sum_{g \in G} hg(v) = w[/tex]

The second equality holds because (I think) an isometry of a Hilbert space is automatically an affine map (it maps affine combinations to affine combinations, and our sum there is an affine combination since the coefficients sum to 1).

The last equality holds because multiplication with a group element is bijective, so it is the same sum, just reordered.
 

1. What is a finite subgroup of isometries of n-dimensional space?

A finite subgroup of isometries of n-dimensional space refers to a collection of isometries (transformations that preserve distances and angles) in n-dimensional space that are finite in number. In other words, there are only a limited number of distinct isometries in the subgroup.

2. How are finite subgroups of isometries of n-dimensional space used in science?

Finite subgroups of isometries of n-dimensional space are widely used in various fields of science, such as physics, chemistry, and engineering. They are particularly useful in studying the properties of crystals and other symmetrical objects, as well as in understanding the behavior of particles in quantum mechanics.

3. Can all isometries in n-dimensional space be grouped into finite subgroups?

No, not all isometries in n-dimensional space can be grouped into finite subgroups. Only those isometries that are finite in number and have a certain level of symmetry can be classified into finite subgroups.

4. What are some examples of finite subgroups of isometries of n-dimensional space?

Some examples of finite subgroups of isometries of n-dimensional space include the symmetry groups of regular polygons and polyhedra, the rotation groups of regular polytopes, and the crystallographic point groups.

5. How do finite subgroups of isometries of n-dimensional space relate to the concept of symmetry?

Finite subgroups of isometries of n-dimensional space are closely related to the concept of symmetry, as they represent the different ways in which an object can be symmetrically arranged in n-dimensional space. They help us understand and classify the symmetries present in various objects and systems, and their study has led to advancements in many scientific fields.

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