Group action

In mathematics, a group action of a group ${\displaystyle G}$ on a set ${\displaystyle S}$ is a group homomorphism from ${\displaystyle G}$ to some group (under function composition) of functions from ${\displaystyle S}$ to itself. It is said that ${\displaystyle G}$ acts on ${\displaystyle S}$.

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 ${\displaystyle \operatorname {GL} (n,K)}$, the group of the invertible matrices of dimension ${\displaystyle n}$ over a field ${\displaystyle K}$.

The symmetric group ${\displaystyle S_{n}}$ acts on any set with ${\displaystyle n}$ 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.

If ${\displaystyle G}$ is a group with identity element ${\displaystyle e}$, and ${\displaystyle X}$ is a set, then a (left) group action ${\displaystyle \alpha }$ of ${\displaystyle G}$ on X is a function

that satisfies the following two axioms:

for all g and h in G and all x in ${\displaystyle X}$.