A group action is a representation of the elements of a group as symmetries of a set. Many groups have a natural group action coming from their construction; e.g. the dihedral group acts on the vertices of a square because the group is given as a set of symmetries of the square. A group action of a group on a set is an abstract generalization of this idea, which can be used to derive useful facts about both the group and the set it acts on.
Formally, a group action of a group on a set is a function satisfying the following properties:
(1) for all
(2) for all and
When the action is clear, the function is often written as With this notation, the axioms become
The standard example of a group action is when equals the symmetric group or a subgroup of and . Then acts on by the formula The properties are clear: when is the identity of and
Here are several basic concepts related to group actions.
Let be a group acting on a set
A fixed point of an element is an element such that
The stabilizer of a point is the set of elements such that is a fixed point of
The orbit of an element is the set of elements such that for some
Let act on by the formulas and
Find the fixed points of every element of
Find the stabilizer of every element of
Find the orbits of the elements of
Every element of is a fixed point of but has exactly one fixed point, namely
The stabilizer of is all of but the stabilizer of any other element of is the trivial subgroup of
The orbit of is and the orbit of any other element is a two-element set
Note that is partitioned as a union of its distinct and disjoint orbits. This is true in general; it follows from the fact that the relation defined by if is in the orbit of is an equivalence relation.
Certain common properties related to these definitions arise in many group actions.
Let be a group acting on a set
The action is transitive if there is only one orbit: for any there is an element such that
The action is faithful if the intersection of the stabilizers for consists only of the trivial element
The axioms of a group action give a group homomorphism where is the group of permutations of the elements of A faithful action, then, is one for which this homomorphism is injective, as its kernel is trivial.
There are many examples where a group acts on objects related to Here are some examples:
(1) Every group acts on itself by left multiplication: acts on via the formula This is a transitive and faithful action; there is one orbit, and in fact the stabilizer of any element is trivial: if and only if is the identity.
(2) Every group acts on itself by conjugation: acts on via the formula The orbits of this action are called conjugacy classes, and the stabilizer of an element is called the centralizer
(3) If is a subgroup of then acts on the set of cosets by left multiplication. The action is transitive, since for any This induces a map whose kernel is equal to the intersection of all the conjugates of which is the largest normal subgroup contained in So if does not contain any nontrivial normal subgroups, the action is faithful. This construction is quite useful in the analysis of simple groups.
Let be the projective complex line: this can be thought of as the set of complex numbers plus the point Then the group of invertible matrices with complex entries acts on via the formula
with the understanding that maps to and maps to If then maps to
Checking that this is a group action boils down to the following computation for the second axiom:
and the matrix is the product which shows that this gives a group action.
Note that the matrix acts trivially on the projective line, since So the action of turns into an action of where is the subgroup of matrices which are constant multiples of the identity. This group is called the projective general linear group the action of is faithful and also transitive. (In fact, it is 3-transitive: any list of three distinct points can be sent to any other list of three distinct points.)
There is a natural relationship between orbits and stabilizers of a group action.
Let be a group acting on a set Fix a point and consider the function given by In fact, this gives a function given by the same formula, where is the stabilizer of Here is the set of left cosets of Note that is not necessarily a normal subgroup. So is just a set of cosets, not a quotient group.
The function is injective, and its image is precisely the orbit of The conclusion is as follows:
There is a bijection between the set and the orbit of
When the orbit of is finite, this gives the following formula:
Let be a group acting on a set Let be the stabilizer of an element Suppose that the orbit of is finite. Then the index is finite and equal to If is finite, then
Let and let Let The orbit of is all of (the action is transitive), so the theorem says that the index is equal to Indeed, is isomorphic to the group of permutations of So and
How many rotational symmetries does a cube have? (Extra credit: what is the group of symmetries of the cube?)
The question is asking for the number of ways to permute the vertices (or faces, or edges) of a cube by twisting and turning (but not reflecting) the cube in 3D-space. The orbit-stabilizer theorem can be used to solve this problem in three different ways.
Let be the group of symmetries of the cube. Then acts on the set of faces of the cube. There are six faces, and the action is transitive, so the size of an orbit of a given face is The order of the stabilizer of a face is 4, because there are four rotations of a cube that fix a face (the rotations around the axis perpendicular to the face). So by the orbit-stabilizer theorem,
But also acts on the vertices of a cube. There are eight vertices, and again the action is transitive. What is the order of the stabilizer of a vertex? The only rotations of a cube that leave one vertex fixed are the three rotations of and degrees around the body diagonal of the cube. So and so by the orbit-stabilizer theorem
And also acts on the edges of a cube. There are twelve edges, and the action is transitive. The stabilizer of an edge has order two; there is the identity of and the unique element of which switches the vertices of the edge. So again
In fact, is isomorphic to the symmetry group and the isomorphism uses yet another action of namely the action of on the body diagonals, of which there are four. This action gives a homomorphism and it is straightforward to show that this homomorphism is injective (the action is faithful), and the groups have the same size, so it must be an isomorphism.