Group action

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This article is about the mathematical concept. For the sociology term, see group action (sociology).

In algebra and geometry, a group action is a description of symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set. In this case, the group is also called a permutation group (especially if the set is finite or not a vector space) or transformation group (especially if the set is a vector space and the group acts like linear transformations of the set).

A group action is an extension to the definition of a symmetry group in which every element of the group "acts" like a bijective transformation (or "symmetry") of some set, without being identified with that transformation. This allows for a more comprehensive description of the symmetries of an object, such as a polyhedron, by allowing the same group to act on several different sets of features, such as the set of vertices, the set of edges and the set of faces of the polyhedron.

If G is a group and X is a set, then a group action may be defined as a group homomorphism h from G to the symmetric group on X. The action assigns a permutation of X to each element of the group in such a way that the permutation of X assigned to:





\varphi : G \times X \to X : (g,x)\mapsto \varphi(g,x)

that satisfies the following two axioms (where we denote φ(g, x) as g.x):[1]

(gh).x = g.(h.x) for all g, h in G and all x in X. (Here, gh denotes the result of applying the group operation of G to the elements g and h.)
e.x = x for all x in X. (Here, e denotes the neutral element of the group G.)

The set X is called a (left) G-set. The group G is said to act on X (on the left).

From these two axioms, it follows that for every g in G, the function which maps x in X to g.x is a bijective map from X to X (its inverse being the function which maps x to g−1.x). Therefore, one may alternatively define a group action of G on X as a group homomorphism from G into the symmetric group Sym(X) of all bijections from X to X.[2]

In complete analogy, one can define a right group action of G on X as an operation X × GX mapping (x, g) to x.g and satisfying the two axioms:

x.(gh) = (x.g).h for all g, h in G and all x in X;
x.e = x for all x in X.



  • The trivial action of any group G on any set X is defined by g.x = x for all g in G and all x in X; that is, every group element induces the identity permutation on X.[3]
  • In every group G, left multiplication is an action of G on G: g.x = gx for all g, x in G.
  • In every group G, conjugation is an action of G on G: g.x = gxg−1. An exponential notation is commonly used for the right-action variant: xg = g−1xg; it satisfies (xg)h = xgh.
  • The sets acted on by a group G comprise the category of G-sets in which the objects are G-sets and the morphisms are G-set homomorphisms: functions f : XY such that (g.f)(x) = f(g.x) for every g in G.
  • The additive group of the real numbers (R, +) acts on the set of real functions of a real vari able in various ways, with (t.f)(x) equal to e.g. f(x + t), f(x) + t, f(xet), f(x)et, f(x + t)et, or f(xet) + t, but not f(xet + t).
  • Given a group action of G on X, we can define an induced action of G on the power set of X, by setting g.U = {g.u : uU} for every subset U of X and every g in G. This is useful, for instance, in studying the action of the large Mathieu group on a 24-set and in studying symmetry in certain models of finite geometries.
  • The quaternions with norm 1 (the versors), as a multiplicative group, act on R3: for any such quaternion \scriptstyle z = \cos\frac{1}{2}\alpha + \sin\frac{1}{2}\alpha \hat{\mathbf{v}}, the mapping f(x) = zxz is a counterclockwise rotation through an angle α about an axis v; z is the same rotation; see quaternions and spatial rotation.

Types of actions[edit]

The action of G on X is called

  • Transitive if X is non-empty and if for any x, y in X there exists a g in G such that g.x = y.
  • Faithful (or effective) if for any two distinct g, h in G there exists an x in X such that g.xh.x; or equivalently, if for any ge in G there exists an x in X such that g.xx. Intuitively, in a faithful group action, different elements of G induce different permutations of X.
  • Free (or semiregular or fixed point free) if, given g, h in G, the existence of an x in X with g.x = h.x implies g = h. Equivalently: if g is a group element and there exists an x in X with g.x = x (that is, if g has at least one fixed point), then g is the identity.
  • Regular (or simply transitive or sharply transitive) if it is both transitive and free; this is equivalent to saying that for any two x, y in X there exists precisely one g in G such that g.x = y. In this case, X is known as a principal homogeneous space for G or as a G-torsor.
  • n-transitive if X has at least n elements and for any pairwise distinct x1, …, xn and pairwise distinct y1, …, yn there is a g in G such that g·xk = yk for 1 ≤ kn. A 2-transitive action is also called doubly transitive, a 3-transitive action is also called triply transitive, and so on. Such actions define 2-transitive groups, 3-transitive groups, and multiply transitive groups.
  • Locally free if G is a topological group, and there is a neighbourhood U of e in G such that the restriction of the action to U is free; that is, if g.x = x for some x and some g in U then g = e.
  • Irreducible if X is a non-zero module over a ring R, the action of G is R-linear, and there is no nonzero proper invariant submodule.

Every free action on a non-empty set is faithful. A group G acts faithfully on X if and only if the corresponding homomorphism G → Sym(X) has a trivial kernel. Thus, for a faithful action, G embeds into to a permutation group on X; specifically, G is isomorphic to its image in Sym(X).


If G does not act faithfully on X, one can easily modify the group to obtain a faithful action. If we define N = {g in G : g.x = x for all x in X}, then N is a normal subgroup of G; indeed, it is the kernel of the homomorphism G → Sym(X). The factor group G/N acts faithfully on X by setting (gN).x = g.x. The original action of G on X is faithful if and only if N = {e}.

Orbits and stabilizers[edit]


Consider a group G acting on a set X. The orbit of a point x in X is the set of elements of X to which x can be moved by the elements of G. The orbit of x is denoted by G.x:

 G.x = \left\{ g.x \mid g \in G \right\}.

The defining properties of a group guarantee that the set of orbits of (points x in) X under the action of G form a partition of X. The associated equivalence relation is defined by saying x ~ y if and only if there exists a g in G with g.x = y. The orbits are then the equivalence classes under this relation; two elements x and y are equivalent if and only if their orbits are the same; i.e., G.x = G.y.

The group action is transitive if and only if it has only one orbit, i.e. if there exists x in X with G.x = X. This is the case if and only if G.x = X for all x in X.


Invariant subsets[edit]

If Y is a subset of X, we write GY for the set { g.y : yY and gG}. We call the subset Y invariant under G if G.Y = Y (which is equivalent to G.YY). In that case, G also operates on Y by restricting the action to Y. The subset Y is called fixed under G if g.y = y for all g in G and all y in Y. Every subset that is fixed under G is also invariant under G, but not vice versa.



Fixed points and stabilizer subgroups[edit]

Given g in G and x in X with g.x = x, we say x is a fixed point of g and g fixes x.

For every x in X, we define the stabilizer subgroup of x (also called the isotropy group) as the set of all elements in G that fix x:

G_x = \{g \in G \mid g.x = x\}.

This is a subgroup of G, though typically not a normal one. The action of G on X is free if and only if all stabilizers are trivial. The kernel N of the homomorphism G → Sym(X) is given by the intersection of the stabilizers Gx for all x in X. If N is trivial, the action is said to be faithful (or effective).

Let x and y be two elements in X, and let g be a group element such that y = g.x. Then the two stabilizer groups Gx and Gy are related by Gy = g Gx g−1. Proof: by definition, hGy if and only if h.(g.x) = g.x. Applying g−1 to both sides of this equality yields (g−1hg).x = (g−1g).x = x; that is, g−1hgGx.

The above says that the stabilizers of elements in the same orbit are conjugate to each other. Thus, to each orbit, one can associate a conjugacy class of a subgroup of G (i.e., the set of all conjugates of the subgroup). Let (H) denote the conjugacy class of H. Then one says that the orbit O has type (H) if the stabilizer G_x of some/any x in O belongs to (H).

Orbit-stabilizer theorem and Burnside's lemma[edit]

Orbits and stabilizers are closely related. For a fixed x in X, consider the map from G to X given by gg.x for all gG. The image of this map is the orbit of x and the coimage is the set of all left cosets of Gx. The standard quotient theorem of set theory then gives a natural bijection between G/Gx and G.x. Specifically, the bijection is given by hGxh.x. This result is known as the orbit-stabilizer theorem. From a more categorical perspective, the orbit-stabilizer theorem comes from the fact that every G-set is a sum of quotients of the G-set G.

If G and X are finite then the orbit-stabilizer theorem, together with Lagrange's theorem, gives

|G.x| = [G\,:\,G_x] = |G| / |G_x|.

This result is especially useful since it can be employed for counting arguments.


\left|X/G\right|=\frac{1}{\left|G\right|}\sum_{g\in G}\left|X^g\right|

where Xg is the set of points fixed by g. This result is mainly of use when G and X are finite, when it can be interpreted as follows: the number of orbits is equal to the average number of points fixed per group element.

Fixing a group G, the set of formal differences of finite G-sets forms a ring called the Burnside ring of G, where addition corresponds to disjoint union, and multiplication to Cartesian product.

Group actions and groupoids[edit]

The notion of group action can be put in a broader context by using the action groupoid \scriptstyle G' \;=\;  G \,\ltimes\, X associated to the group action, thus allowing techniques from groupoid theory such as presentations and fibrations. Further the stabilisers of the action are the vertex groups, and the orbits of the action are the components, of the action groupoid. For more details, see the book Topology and groupoids referenced below.

This action groupoid comes with a morphism \scriptstyle p:\; G' \,\rightarrow\, G which is a covering morphism of groupoids. This allows a relation between such morphisms and covering maps in topology.

Morphisms and isomorphisms between G-sets[edit]

If X and Y are two G-sets, we define a morphism from X to Y to be a function f : XY such that f(g.x) = g.f(x) for all g in G and all x in X. Morphisms of G-sets are also called equivariant maps or G-maps.

The composition of two morphisms is again a morphism.


Some example isomorphisms:



Continuous group actions[edit]


If G is a discrete group acting on a topological space X, the action is properly discontinuous if for any point x in X there is an open neighborhood U of x in X, such that the set of all g in G for which \scriptstyle g(U) \,\cap\, U \;\ne\; \emptyset consists of the identity only. If X is a regular covering space of another topological space Y, then the action of the deck transformation group on X is properly discontinuous as well as being free. Every free, properly discontinuous action of a group G on a path-connected topological space X arises in this manner: the quotient map XX/G is a regular covering map, and the deck transformation group is the given action of G on X. Furthermore, if X is simply connected, the fundamental group of X/G will be isomorphic to G.



The action of G on X is said to be proper if the mapping G × XX × X that sends (g, x) ↦ (g.x, x) is a proper map.

Strongly continuous group action and smooth points[edit]

A group action of a topological group G on a topological space X is said to be strongly continuous if for all x in X, the map gg.x is continuous with respect to the respective topologies. Such an action induces an action on the space of continuous functions on X by defining (g.f)(x) = f(g−1.x) for every g in G, f a continuous function on X, and x in X. Note that, while every continuous group action is strongly continuous, the converse is not in general true.[4]

The subspace of smooth points for the action is the subspace of X of points x such that gg.x is smooth; i.e., it is continuous and all derivatives[where?] are continuous.

Variants and generalizations[edit]



One can view a group G as a category with a single object in which every morphism is invertible. A group action is then nothing but a functor from G to the category of sets, and a group representation is a functor from G to the category of vector spaces. A morphism between G-sets is then a natural transformation between the group action functors. In analogy, an action of a groupoid is a functor from the groupoid to the category of sets or to some other category.


See also[edit]


  2. ^ This is done e.g. by Smith (2008). Introduction to abstract algebra. p. 253. 


External links[edit]