Group action

also called Transformation group and has 19 more synonyms ↓

• Action
• Action of a group
• Faithful group action
• G-set
• Group
• Group actions
• Group of transformation
• Group Orbit
• Group transformation
• Groups acting on sets
• Isotropy group
• Isotropy subgroup
• Multiply transitive
• Orbit space
• Orbit stabiliser theorem
• Orbit-stabiliser theorem
• Orbit-Stabilizer theorem
• Permutation representation
• Stabilizer subgroup
• Suggest another synonym or correct a wrong one

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1. In algebra and geometry, a group action is a way of describing symmetries of objects using groups

   

   created by factobot on November 12, 2008

2. The essential elements of the object are described by a set and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set

   

   created by factobot on November 12, 2008

4. In this case, the group is also called a permutation group (especially if the set is finite or not a vector space) or transformation group (especially if the set is a vector space and the group acts like linear transformations of the set)

   

   created by factobot on November 12, 2008

5. a flexible generalization of the notion of a symmetry group in which every element of the group "acts" like bijective transformation (or "symmetry") of some set, without being identified with that transformation

   

   created by factobot on November 12, 2008

6. This allows for a more comprehensive description of the symmetries of an object, such as a polyhedron, by allowing the same group to act on several different sets, such as the set of vertices, the set of edges and the set of faces of the polyhedron

   

   created by factobot on November 12, 2008

7. 

   Given an equilateral triangle, the counterclockwise rotation by 120° around the center of the triangle "acts" on the set of vertices of the triangle by mapping every vertex to another one

   

   created by factobot on November 12, 2008

8. If G is a group and X is a set, then a (left) group action of G on X is a binary function denoted which satisfies the following two axioms

   definition

   created by factobot on November 12, 2008

9. The action of G on X is called transitive if for any two x, y in X there exists a g in G such that g·x = y

   action type

   created by factobot on November 12, 2008

10. Consider a group G acting on a set X. The orbit of a point x in X is the set of elements of X to which x can be moved by the elements of G. The orbit of x is denoted by Gx

    orbit stabilizer

    created by factobot on November 12, 2008

11. The defining properties of a group guarantee that the set of orbits of X under the action of G form a partition of X. The associated equivalence relation is defined by saying x ~ y if and only if there exists a g in G with g·x = y

    orbit stabilizer

    created by factobot on November 12, 2008

12. The notion of group action can be put in a broader context by using the associated `action groupoid' G' = G\ltimes X associated to the group action, thus allowing techniques from groupoid theory such as presentations and fibrations

    group groupoid

    created by factobot on November 12, 2008

13. If X and Y are two G-sets, we define a morphism from X to Y to be a function f : X → Y such that f(g.x) = g.f(x) for all g in G and all x in X. Morphisms of G-sets are also called equivariant maps or G-maps

    isomorphism morphism

    created by factobot on November 12, 2008

14. One often considers continuous group actions: the group G is a topological group, X is a topological space, and the map G ×

    continuous group

    created by factobot on November 12, 2008

15. One can also consider actions of monoids on sets, by using the same two axioms as above

    generalization

    created by factobot on November 12, 2008