Group action

In algebra and geometry, a group action is a way of describing 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 from G to the symmetric group of 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:

The identity element of G is the identity transformation of X;
A product gh of two elements of G is the composite of the permutations assigned to g and h.

Since each element of G is represented as a permutation, a group action is also known as a permutation representation.

The abstraction provided by group actions is a powerful one, because it allows geometrical ideas to be applied to more abstract objects. Many objects in mathematics have natural group actions defined on them. In particular, groups can act on other groups, or even on themselves. Despite this generality, the theory of group actions contains wide-reaching theorems, such as the orbit stabilizer theorem, which can be used to prove deep results in several fields.

1 Definition
2 Examples
3 Types of actions
4 Orbits and stabilizers
5 Group actions and groupoids
6 Morphisms and isomorphisms between G-sets
7 Continuous group actions
- 7.1 Strongly continuous group action and smooth points
8 Generalizations
9 See also
10 Notes
11 References

Definition

If $\scriptstyle\left(G,\, \cdot\right)$ is a group and $\scriptstyle X$ is a set, then a (left) group action of G on X is a binary operator:

$\circ: G \times X \to X$

that satisfies the following two axioms:^[1]

Associativity: $\left(g \,\cdot\, h\right) \,\circ\, x \;=\; g \,\circ\, \left(h \,\circ\, x\right),\ \forall g,\, h \,\in\, G,\; x \,\in\, X$ ;
Identity: $e \,\circ\, x \;=\; x,\ \forall x \,\in\, X$ .

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 a function X × G → X by the two axioms:

Associativity: $x \,\circ\, \left(g \,\cdot\, h\right) \;=\; \left(x \,\circ\, g\right) \,\circ\, h ,\ \forall g,\, h \,\in\, G,\; x \,\in\, X$ ;
Identity: $x \,\circ\, e \;=\; x,\ \forall x \,\in\, X$ .

The difference between left and right actions is in the order in which a product like gh acts on x. For a left action h acts first and is followed by g, while for a right action g acts first and is followed by h. From a right action a left action can be constructed by composing with the inverse operation on the group. If $\scriptstyle \circ_r$ is a right action, then, a the following is a left action:

$\circ_l: G \times X \to X,\ g \circ_l x \mapsto x \circ_r g^{-1}$

Satisfying associativity,

$\begin{align} (g \cdot h) \circ_l x &= x \circ_r (g \cdot h)^{-1} = x \circ_r (h^{-1} \cdot g^{-1})\\ &= (x \circ_r h^{-1}) \circ_r g^{-1} = (h \circ_l x) \circ_r g^{-1}\\ &= g \circ_l (h \circ_l x) \end{align}$

and identity:

$e \circ_l x = x \circ_r e^{-1} = x \circ_r e = x$

Any right action has an equivalent left action, thus only left actions can be considered without any loss of generality. Also, a right action of a group $G$ on $X$ is the same thing as a left action of its opposite group $G^{op}$ on $X$ .

Examples

The trivial action for any group G is defined by g·x=x for all g in G and all x in X; that is, the whole group G induces the identity permutation on X.^[3]
Every group G acts on G in two natural but essentially different ways: g·x = gx for all x in G, or g·x = gxg⁻¹ for all x in G. The latter action is often called the conjugation action, and an exponential notation is commonly used for the right-action variant: x^g = g⁻¹xg; it satisfies (x^g)^h = x^gh.
The symmetric group S_n and its subgroups act on the set { 1, … , n } by permuting its elements
The symmetry group of a polyhedron acts on the set of vertices of that polyhedron. It also acts on the set of faces or the set of edges of the polyhedron.
The symmetry group of any geometrical object acts on the set of points of that object
The automorphism group of a vector space (or graph, or group, or ring...) acts on the vector space (or set of vertices of the graph, or group, or ring...).
The general linear group GL(n, R), special linear group SL(n, R), orthogonal group O(n, R), and special orthogonal group SO(n, R) are Lie groups which act on Rⁿ.
The Galois group of a field extension E/F acts on the bigger field E. So does every subgroup of the Galois group.
The additive group of the real numbers (R, +) acts on the phase space of "well-behaved" systems in classical mechanics (and in more general dynamical systems): if t is in R and x is in the phase space, then x describes a state of the system, and t·x is defined to be the state of the system t seconds later if t is positive or −t seconds ago if t is negative.
The additive group of the real numbers (R, +) acts on the set of real functions of a real variable with (g·f)(x) equal to e.g. f(x + g), f(x) + g, $\scriptstyle f(x e^g)$ , $\scriptstyle f(x) e^g$ , $\scriptstyle f(x \,%2B\, g) e^g$ , or $\scriptstyle f(x e^g) \,%2B\, g$ , but not $\scriptstyle f(x e^g \,%2B\, g)$
The quaternions with modulus 1, as a multiplicative group, act on R³: for any such quaternion $\scriptstyle z \;=\; \cos\frac{1}{2}\alpha \,%2B\, \sin\frac{1}{2}\alpha\, \hat{\mathbf{v}}$ , the mapping f(x) = z x z^* is a counterclockwise rotation through an angle $\scriptstyle\alpha\,$ about an axis v; −z is the same rotation; see quaternions and spatial rotation.
The isometries of the plane act on the set of 2D images and patterns, such as a wallpaper pattern. The definition can be made more precise by specifying what is meant by image or pattern; e.g., a function of position with values in a set of colors.
More generally, a group of bijections g: V → V acts on the set of functions x: V → W by (gx)(v) = x(g⁻¹(v)) (or a restricted set of such functions that is closed under the group action). Thus a group of bijections of space induces a group action on "objects" in it.

Types of actions

The action of G on X is called

Transitive if X is non-empty and if equivalently
1. For any x, y in X there exists a g in G such that gx = y,
2. Gx = X for all x in X,
3. Gx = X for some x in X.
Here, Gx = {g.x | g in G} is the orbit of x under G.
- Sharply transitive if that g is unique; it is equivalent to regularity defined below.
n-transitive if X has at least n elements and for any pairwise distinct x₁, ..., x_n and pairwise distinct y₁, ..., y_n there is a g in G such that g.x_k = y_k for 1 ≤ k ≤ n. 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.
- Sharply n-transitive if there is exactly one such g. See also sharply triply transitive groups.
Faithful (or effective) if for any two distinct g, h in G there exists an x in X such that g·x ≠ h·x; or equivalently, if for any g≠ e in G there exists an x in X such that g·x ≠ x. Intuitively, different elements of G induce different permutations of X.
Free (or semiregular) if for any x in X, g.x = h.x implies g = h. Equivalently: if there exists an x in X such that g.x = x (that is, if g has at least one fixed point), then g is the identity.
Regular (or simply 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.
Primitive if it is transitive and preserves no non-trivial partition of X. See Primitive permutation group for details.
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 homomorphism G → Sym(X) has a trivial kernel. Thus, for a faithful action, G is isomorphic to a permutation group on X; specifically, G is isomorphic to its image in Sym(X).

The action of any group G on itself by left multiplication is regular, and thus faithful as well. Every group can, therefore, be embedded in the symmetric group on its own elements, Sym(G) — a result known as Cayley's theorem.

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

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 Gx:

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

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., Gx = Gy.

The set of all orbits of X under the action of G is written as X /G (or, less frequently: G \X), and is called the quotient of the action. In geometric situations it may be called the orbit space, while in algebraic situations it may be called the space of coinvariants, and written $\scriptstyle X_G,$ by contrast with the invariants (fixed points), denoted $\scriptstyle X^G:$ the coinvariants are a quotient while the invariants are a subset. The coinvariant terminology and notation are used particularly in group cohomology and group homology, which use the same superscript/subscript convention.

Invariant subsets

If Y is a subset of X, we write GY for the set { g·y : y ∈ Y and g ∈ G}. We call the subset Y invariant under G if GY = Y (which is equivalent to GY ⊆ Y). In that case, G also operates on 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's fixed under G is also invariant under G, but not vice versa.

Every orbit is an invariant subset of X on which G acts transitively. The action of G on X is transitive if and only if all elements are equivalent, meaning that there is only one orbit.

Stabilizer subgroup

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

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

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 G_x for all x in X.

A useful result is the following. Let x and y be two distinct elements in X, and let g be a group element such that $\scriptstyle y \;=\; g \,\cdot\, x$ . Then the two isotropy groups $\scriptstyle G_x$ and $\scriptstyle G_y$ are related by $\scriptstyle G_y \;=\; g G_x g^{-1}$ . Let us prove this: by definition $\scriptstyle h \,\in\, G_y$ if and only if $\scriptstyle h \,\cdot\, (g \,\cdot\, x) = g \,\cdot\, x$ . Applying $\scriptstyle g^{-1}$ to both sides of this equality we get $\scriptstyle g^{-1} \,\cdot\, h \,\cdot\, g \,\cdot\, x \;=\; g^{-1} \,\cdot\, g \,\cdot\, x \;=\; x$ ; that is, $\scriptstyle g^{-1}hg \,\in\, G_x$ . This shows that $\scriptstyle h \,\in\, G_y$ if and only if $\scriptstyle h \,\in\, gG_x g^{-1}$ .

Orbit-stabilizer theorem

Orbits and stabilizers are closely related. For a fixed x in X, consider the map from G to X given by g ↦ g·x for all g ∈ G. The image of this map is the orbit of x and the coimage is the set of all left cosets of G_x. The standard quotient theorem of set theory then gives a natural bijection between G /G_x and Gx. Specifically, the bijection is given by hG_x ↦ h·x. This result is known as the orbit-stabilizer theorem.

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

$|Gx| = [G\,:\,G_x] = |G| / |G_x|.$

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

Note that if two elements x and y belong to the same orbit, then their stabilizer subgroups, G_x and G_y, are conjugate (in particular, they are isomorphic). More precisely: if y = g·x, then G_y = gG_x g⁻¹. Points with conjugate stabilizer subgroups are said to have the same orbit-type.

A result closely related to the orbit-stabilizer theorem is Burnside's lemma:

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

where X^g 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.

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

A G-invariant element of X is x ∈ X such that g·x = x for all g ∈ G. The set of all such x is denoted X^G and called the G-invariants of X. When X is a G-module, X^G is the zeroth group cohomology group of G with coefficients in X, and the higher cohomology groups are the derived functors of the functor of G-invariants.

Group actions and groupoids

The notion of group action can be put in a broader context by using the associated 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

If X and Y are two G-sets, we define a morphism from X to Y to be a function f : X → Y 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.

If such a function f is bijective, then its inverse is also a morphism, and we call f an isomorphism and the two G-sets X and Y are called isomorphic; for all practical purposes, they are indistinguishable in this case.

Some example isomorphisms:

Every regular G action is isomorphic to the action of G on G given by left multiplication.
Every free G action is isomorphic to G×S, where S is some set and G acts by left multiplication on the first coordinate.
Every transitive G action is isomorphic to left multiplication by G on the set of left cosets of some subgroup H of G.

With this notion of morphism, the collection of all G-sets forms a category; this category is a Grothendieck topos (in fact, assuming a classical metalogic, this topos will even be Boolean).

Continuous group actions

One often considers continuous group actions: the group G is a topological group, X is a topological space, and the map G × X → X is continuous with respect to the product topology of G × X. The space X is also called a G-space in this case. This is indeed a generalization, since every group can be considered a topological group by using the discrete topology. All the concepts introduced above still work in this context, however we define morphisms between G-spaces to be continuous maps compatible with the action of G. The quotient X/G inherits the quotient topology from X, and is called the quotient space of the action. The above statements about isomorphisms for regular, free and transitive actions are no longer valid for continuous group actions.

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 $\scriptstyle g \,\in\, G$ 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 X ↦ X/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 $\scriptstyle X/G$ will be isomorphic to $\scriptstyle G$ .

These results have been generalised in the book Topology and Groupoids referenced below to obtain the fundamental groupoid of the orbit space of a discontinuous action of a discrete group on a Hausdorff space, as, under reasonable local conditions, the orbit groupoid of the fundamental groupoid of the space. This allows calculations such as the fundamental group of the symmetric square of a space X, namely the orbit space of the product of X with itself under the twist action of the cyclic group of order 2 sending (x,y) to (y,x).

An action of a group G on a locally compact space X is cocompact if there exists a compact subset A of X such that GA = X. For a properly discontinuous action, cocompactness is equivalent to compactness of the quotient space X/G.

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

Strongly continuous group action and smooth points

If $\scriptstyle \alpha:\; G\times X \,\to\, X$ is an action of a topological group $\scriptstyle G$ on another topological space $\scriptstyle X$ , one says that it is strongly continuous if for all $\scriptstyle x \,\in\, X$ , the map g ↦ α_g(x) is continuous with respect to the respective topologies. Such an action induces an action on the space of continuous function on $\scriptstyle X$ by $\scriptstyle (\alpha_gf)(x) \;=\; f(\alpha_g^{-1}x)$ .

The subspace of smooth points for the action $\alpha$ is the subspace of $X$ of points $x$ such that g ↦ α_g(x) is smooth; i.e., it is continuous and all derivatives are continuous.

Generalizations

One can also consider actions of monoids on sets, by using the same two axioms as above. This does not define bijective maps and equivalence relations however. See semigroup action.

Instead of actions on sets, one can define actions of groups and monoids on objects of an arbitrary category: start with an object X of some category, and then define an action on X as a monoid homomorphism into the monoid of endomorphisms of X. If X has an underlying set, then all definitions and facts stated above can be carried over. For example, if we take the category of vector spaces, we obtain group representations in this fashion.

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.

Without using the language of categories, one can extend the notion of a group action on a set X by studying as well its induced action on the power set of X. 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.

Notes

^ Eie & Chang (2010), p. 144
^ This is done e.g. by Smith (2008), p. 253
^ Eie & Chang (2010), p. 145

References

Aschbacher, Michael (2000). Finite Group Theory. Cambridge University Press. ISBN 978-0-521-78675-1. MR 1777008
Brown, Ronald (2006). Topology and groupoids, Booksurge PLC, ISBN 1-4196-2722-8.
Categories and groupoids, P.J. Higgins, downloadable reprint of van Nostrand Notes in Mathematics, 1971, which deal with applications of groupoids in group theory and topology.
Dummit, David; Richard Foote (2003). Abstract Algebra ((3rd ed.) ed.). Wiley. ISBN 0-471-43334-9.
Rotman, Joseph (1995). An Introduction to the Theory of Groups. Graduate Texts in Mathematics 148 ((4th ed.) ed.). Springer-Verlag. ISBN 0-387-94285-8.
Smith, Jonathan D.H. (2008). Introduction to abstract algebra. Textbooks in mathematics. CRC Press. ISBN 9781420063714.
Eie, Minking; Chang, Shou-Te (2010). A Course on Abstract Algebra. World Scientific. ISBN 9789814271882.
Weisstein, Eric W., "Group Action" from MathWorld.