## ::Group action

### ::concepts

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In mathematics, a symmetry group is an abstraction used to describe the symmetries of an object. A **group action** formalizes the relationship between the group and the symmetries of the object. It relates each element of the group to a particular transformation of the object.

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). A **permutation representation** of a group *G* is a representation of *G* as a group of permutations of the set (usually if the set is finite), and may be described as a group representation of *G* by permutation matrices. It is the same as a group action of *G* on an *ordered* basis of a vector space.

A group action is an extension to the notion of a symmetry group in which every element of the group "acts" like a bijective transformation (or "symmetry") of some set, without being identified with that transformation. 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 of features, such as the set of vertices, the set of edges and the set of faces of the polyhedron.

If *G* is a group and *X* is a set, then a group action may be defined as a group homomorphism *h* from *G* to the symmetric group on *X*. The action assigns a permutation of *X* to each element of the group in such a way that the permutation of *X* assigned to

- the identity element of
*G*is the identity transformation of*X*; - a product
*gk*of two elements of*G*is the composition of the permutations assigned to*g*and*k*.

The abstraction provided by group actions is a powerful one, because it allows geometrical ideas to be applied to more abstract objects. Many objects in mathematics have natural group actions defined on them. In particular, groups can act on other groups, or even on themselves. Despite this generality, the theory of group actions contains wide-reaching theorems, such as the orbit stabilizer theorem, which can be used to prove deep results in several fields.

**Group action sections**

Intro Definition Examples Group actions and groupoids Morphisms and isomorphisms between

*G*-sets Continuous group actions Variants and generalizations See also Notes References External links

PREVIOUS: Intro | NEXT: Definition |

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Group::''g'' ''X''::''x'' ''g''::action Nowrap::space Actions::''f'' ''h''::''y''

{{ safesubst:#invoke:Unsubst||$N=Technical |date=__DATE__ |$B=
{{#invoke:Message box|ambox}}
}}
{{#invoke:Hatnote|hatnote}}

In mathematics, a symmetry group is an abstraction used to describe the symmetries of an object. A **group action** formalizes the relationship between the group and the symmetries of the object. It relates each element of the group to a particular transformation of the object.

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). A **permutation representation** of a group *G* is a representation of *G* as a group of permutations of the set (usually if the set is finite), and may be described as a group representation of *G* by permutation matrices. It is the same as a group action of *G* on an *ordered* basis of a vector space.

A group action is an extension to the notion of a symmetry group in which every element of the group "acts" like a bijective transformation (or "symmetry") of some set, without being identified with that transformation. 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 of features, such as the set of vertices, the set of edges and the set of faces of the polyhedron.

If *G* is a group and *X* is a set, then a group action may be defined as a group homomorphism *h* from *G* to the symmetric group on *X*. The action assigns a permutation of *X* to each element of the group in such a way that the permutation of *X* assigned to

- the identity element of
*G*is the identity transformation of*X*; - a product
*gk*of two elements of*G*is the composition of the permutations assigned to*g*and*k*.

The abstraction provided by group actions is a powerful one, because it allows geometrical ideas to be applied to more abstract objects. Many objects in mathematics have natural group actions defined on them. In particular, groups can act on other groups, or even on themselves. Despite this generality, the theory of group actions contains wide-reaching theorems, such as the orbit stabilizer theorem, which can be used to prove deep results in several fields.

**Group action sections**

Intro Definition Examples Group actions and groupoids Morphisms and isomorphisms between

*G*-sets Continuous group actions Variants and generalizations See also Notes References External links

PREVIOUS: Intro | NEXT: Definition |

<< | >> |