Editing Group action (section)

{{Short description|Transformations induced by a mathematical group}}
{{About|the mathematical concept|the sociology term|group action (sociology)}}
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[[File:Group action on equilateral triangle.svg|right|thumb|The [[cyclic group]] {{math|C<sub>3</sub>}} consisting of the [[Rotation (mathematics)|rotations]] by 0°, 120° and 240° acts on the set of the three vertices.]]

In [[mathematics]], a '''group action''' of a group <math>G</math> on a [[set (mathematics)|set]] <math>S</math> is a [[group homomorphism]] from <math>G</math> to some group (under [[function composition]]) of functions from <math>S</math> to itself. It is said that <math>G</math> '''acts''' on <math>S</math>.

Many sets of [[transformation (function)|transformation]]s form a [[group (mathematics)|group]] under [[function composition]]; for example, the [[rotation (mathematics)|rotation]]s around a point in the plane. It is often useful to consider the group as an [[abstract group]], and to say that one has a group action of the abstract group that consists of performing the transformations of the group of transformations. The reason for distinguishing the group from the transformations is that, generally, a group of transformations of a [[mathematical structure|structure]] acts also on various related structures; for example, the above rotation group also acts on triangles by transforming triangles into triangles.

If a group acts on a structure, it will usually also act on objects built from that structure. For example, the group of [[Euclidean isometry|Euclidean isometries]] acts on [[Euclidean space]] and also on the figures drawn in it; in particular, it acts on the set of all [[triangle]]s. Similarly, the group of [[symmetries]] of a [[polyhedron]] acts on the [[vertex (geometry)|vertices]], the [[edge (geometry)|edges]], and the [[face (geometry)|faces]] of the polyhedron.

A group action on a [[vector space]] is called a [[Group representation|representation]] of the group. In the case of a finite-dimensional vector space, it allows one to identify many groups with [[subgroups]] of the [[general linear group]] <math>\operatorname{GL}(n,K)</math>, the group of the [[invertible matrices]] of [[dimension]] <math>n</math> over a [[Field (mathematics)|field]] <math>K</math>.

The [[symmetric group]] <math>S_n</math> acts on any [[set (mathematics)|set]] with <math>n</math> elements by permuting the elements of the set. Although the group of all [[permutation]]s of a set depends formally on the set, the concept of group action allows one to consider a single group for studying the permutations of all sets with the same [[cardinality]].