Group action explained
In mathematics, an action of a group
on a
set
is, loosely speaking, an operation that takes an element of
and an element of
and produces another element of
More formally, it is a
group homomorphism from
to the
automorphism group of
(the set of all
bijections on
along with group operation being
function composition). One says that
acts on
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.
, the group of the
invertible matrices of
dimension
over a
field
.
acts on any
set with
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.
Definition
Left group action
If
is a
group with
identity element
, and
is a set, then a (
left)
group action
of
on
is a
function
that satisfies the following two
axioms:
[1] | Identity: |
|
| Compatibility: | \alpha(g,\alpha(h,x))=\alpha(gh,x)
| |
for all
and
in
and all
in
.
The group
is then said to act on
(from the left). A set
together with an action of
is called a (
left)
-
set.
It can be notationally convenient to curry the action
, so that, instead, one has a collection of
transformations
, with one transformation
for each group element
. The identity and compatibility relations then read
and
The second axiom states that the
function composition is compatible with the group multiplication; they form a
commutative diagram. This axiom can be shortened even further, and written as
\alphag\circ\alphah=\alphagh
.
With the above understanding, it is very common to avoid writing
entirely, and to replace it with either a dot, or with nothing at all. Thus,
can be shortened to
or
, especially when the action is clear from context. The axioms are then
