Group actions (Mathematics)

In mathematics, an action of a group G {\displaystyle G} on a set S {\displaystyle S} is, loosely speaking, an operation that takes an element of G {\displaystyle G} and an element of S {\displaystyle S} and produces another element of S . {\displaystyle S.} More formally, it is a group homomorphism from G {\displaystyle G} to the automorphism group of S {\displaystyle S} (the set of all bijections on S {\displaystyle S} along with group operation being function composition). One says that G {\displaystyle G} acts on S . {\displaystyle S.} Many sets of transformations form a group under function composition; for example, the rotations 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 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 isometries acts on Euclidean space and also on the figures drawn in it; in particular, it acts on the set of all triangles. Similarly, the group of symmetries of a polyhedron acts on the vertices, the edges, and the faces of the polyhedron. A group action on a vector space is called a 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 GL ⁡ ( n , K ) {\displaystyle \operatorname {GL} (n,K)} , the group of the invertible matrices of dimension n {\displaystyle n} over a field K {\displaystyle K} . The symmetric group S n {\displaystyle S_{n}} acts on any set with n {\displaystyle n} elements by permuting the elements of the set. Although the group of all permutations 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.