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