Group action and equivalence relation

In summary, given a group G acting on a set X, we obtain an equivalence relation R on X where xRy if and only if x is in the orbit of y. The question is whether for any set X with an equivalence relation R, there exists a group G with an action over X such that the set of its orbits is the same as the equivalence classes of R. It has been proven that this is possible by taking the group of bijections from X to itself and considering the subgroup that leaves the equivalence classes invariant. This shows that every equivalence relation can be seen as the result of a group action, allowing for a better understanding of quotient spaces.
  • #1
Damidami
94
0
Given a group [itex]G[/itex] acting on a set [itex]X[/itex] we get an equivalence relation [itex]R[/itex] on [itex]X[/itex] by [itex]xRy[/itex] iff [itex]x[/itex] is in the orbit of [itex]y[/itex].

My question is, does some form of "reciprocal" always work in the following sense: given a set [itex]X[/itex] with an equivalence relation [itex]R[/itex] defined on it, does it always exist some group [itex]G[/itex] with some action over [itex]X[/itex] such that the set of its orbits coincide with the equivalence classes?

I have thoght it, and concluded that for finite sets and groups, the cardinal of [itex]G[/itex] has to be a múltiple of the cardinal of every orbit, but I can't see if it is always possible to construct such group with such an action.

Thanks in advance for any help!
 
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  • #2
An equivalence relation is the same as a partition of the set into a disjoint union of subsets (the equivalence classes). Let G be the group of bijections from X to itself. Let H be the subset of G which leaves the equivalence classes invariant. Then H is a subgroup, and it acts in the way you want.
 
  • #3
Hi Vargo,
Thanks for your reply. I think I can see your point.
By the subset of G which leaves the equivalence classes invariant, I think you mean the maximal one with that property (as the trivial susbset of G obviously leaves the classes invariant)
Anyway it's interesting that any equivalence relation can be thought as the result of a group action, so every time I see a quotient space of any kind I can think as the result of some group acting by "gluing" some elements together.
 

1. What is a group action and how is it related to equivalence relations?

A group action is a mathematical concept that describes the way a group of objects can act on another set of objects. This action can be thought of as a transformation of the objects in the second set. An equivalence relation is a relation between elements of a set that groups them together based on some common property. Group actions can be used to define equivalence relations by considering the transformations that preserve this relation between elements.

2. Why are group actions important in mathematics?

Group actions are important because they help us understand and solve problems in various areas of mathematics, such as abstract algebra, geometry, and topology. They provide a powerful tool for studying symmetry and invariance in mathematical structures and can be used to prove important theorems and conjectures.

3. How can group actions be visualized?

Group actions can be visualized through various means, such as geometric transformations, permutation puzzles, and symmetrical patterns. For example, a group action on a set of points in the plane can be visualized as a transformation of those points, such as rotations, reflections, and translations. In permutation puzzles, the action of shuffling the pieces represents a group action on the set of all possible arrangements. Symmetrical patterns, such as tessellations, also illustrate group actions.

4. What is the connection between group actions and symmetry?

Group actions and symmetry are closely related concepts. In fact, symmetry can be thought of as a special case of group actions, where the group is the set of all possible transformations that preserve a particular property of an object. Group actions allow us to formally define and study different types of symmetry, such as rotational symmetry, translational symmetry, and reflectional symmetry.

5. Can you give an example of a group action and its corresponding equivalence relation?

One example of a group action and its associated equivalence relation is the action of rotations on a square. The group of rotations is the set of all possible rotations of the square, while the equivalence relation is defined by considering two points on the square as equivalent if they are related by a rotation. This action defines four equivalence classes, each representing a different rotation of the square.

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