Group action

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.