In mathematics, an action of a group G {\displaystyle G} on a set S {\displaystyle S} is, loosely speaking, an operation that takes an element of G {\displaystyle G} and an element of S {\displaystyle S} and produces another element of S . {\displaystyle S.} More formally, it is a group homomorphism from G {\displaystyle G} to the automorphism group of S {\displaystyle S} (the set of all bijections on S {\displaystyle S} along with group operation being function composition). One says 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.
Definition
Left group action If G {\displaystyle G} is a group with identity element e {\displaystyle e} , and X {\displaystyle X} is a set, then a (left) group action α {\displaystyle \alpha } of G {\displaystyle G} on X {\displaystyle X} is a function
α : G × X → X {\displaystyle \alpha :G\times X\to X}
that satisfies the following two axioms:
for all g {\displaystyle g} and h {\displaystyle h} in G {\displaystyle G} and all x {\displaystyle x} in X {\displaystyle X} . The group G {\displaystyle G} is then said to act on X {\displaystyle X} (from the left). A set X {\displaystyle X} together with an action of G {\displaystyle G} is called a (left) G {\displaystyle G} -set. It can be notationally convenient to curry the action α {\displaystyle \alpha } , so that, instead, one has a collection of transformations α g : X → X {\displaystyle \alpha _{g}:X\rightarrow X} , with one transformation α g {\displaystyle \alpha _{g}} for each group element g ∈ G {\displaystyle g\in G} . The identity and compatibility relations then read
α e ( x ) = x {\displaystyle \alpha _{e}(x)=x}
and
α g ( α h ( x ) ) = ( α g ∘ α h ) ( x ) = α g h ( x ) {\displaystyle \alpha _{g}(\alpha _{h}(x))=(\alpha _{g}\circ \alpha _{h})(x)=\alpha _{gh}(x)}
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 α g ∘ α h = α g h {\displaystyle \alpha _{g}\circ \alpha _{h}=\alpha _{gh}} . With the above understanding, it is very common to avoid writing α {\displaystyle \alpha } entirely, and to replace it with either a dot, or with nothing at all. Thus, α ( g , x ) {\displaystyle \alpha (g,x)} can be shortened to g ⋅ x {\displaystyle g\cdot x} or g x {\displaystyle gx} , especially when the action is clear from context. The axioms are then
{ e ⋅ x = x g ⋅ ( h ⋅ x ) = ( g h ) ⋅ x {\displaystyle \left\{{\begin{aligned}&e\cdot x=x\\&g\cdot (h\cdot x)=(gh)\cdot x\end{aligned}}\right.}
From these two axioms, it follows that for any fixed g {\displaystyle g} in G {\displaystyle G} , the function from X {\displaystyle X} to itself which maps x {\displaystyle x} to g ⋅ x {\displaystyle g\cdot x} is a bijection, with inverse bijection the corresponding map for g − 1 {\displaystyle g^{-1}} . Therefore, one may equivalently define a group action of G {\displaystyle G} on X {\displaystyle X} as a group homomorphism from G {\displaystyle G} into the symmetric group Sym ( X ) {\displaystyle \operatorname {Sym} (X)} of all bijections from X {\displaystyle X} to itself.
Right group action Likewise, a right group action of G {\displaystyle G} on X {\displaystyle X} is a function
α : X × G → X , {\displaystyle \alpha :X\times G\to X,}
that satisfies the analogous axioms:
(with α ( x , g ) {\displaystyle \alpha (x,g)} often shortened to x g {\displaystyle xg} or x ⋅ g {\displaystyle x\cdot g} when the action being considered is clear from context)
for all g {\displaystyle g} and h {\displaystyle h} in G {\displaystyle G} and all x {\displaystyle x} in X {\displaystyle X} . The difference between left and right actions is in the order in which a product g h {\displaystyle gh} acts on x {\displaystyle x} . For a left action, h {\displaystyle h} acts first, followed by g {\displaystyle g} second. For a right action, g {\displaystyle g} acts first, followed by h {\displaystyle h} second. Because of the formula ( g h ) − 1 = h − 1 g − 1 {\displaystyle (gh)^{-1}=h^{-1}g^{-1}} , a left action can be constructed from a right action by composing with the inverse operation of the group. Also, a right action of a group G {\displaystyle G} on X {\displaystyle X} can be considered as a left action of its opposite group G op {\displaystyle G^{\text{op}}} on X {\displaystyle X} . Thus, for establishing general properties of a single group action, it suffices to consider only left actions.
Notable properties of actions Let G {\displaystyle G} be a group acting on a set X {\displaystyle X} . The action is called faithful or effective if g ⋅ x = x {\displaystyle g\cdot x=x} for all x ∈ X {\displaystyle x\in X} implies that g = e G {\displaystyle g=e_{G}} . Equivalently, the homomorphism from G {\displaystyle G} to the group of bijections of X {\displaystyle X} corresponding to the action is injective. The action is called free (or semiregular or fixed-point free) if the statement that g ⋅ x = x {\displaystyle g\cdot x=x} for some x ∈ X {\displaystyle x\in X} already implies that g = e G {\displaystyle g=e_{G}} . In other words, no non-trivial element of G {\displaystyle G} fixes a point of X {\displaystyle X} . This is a much stronger property than faithfulness. For example, the action of any group on itself by left multiplication is free. This observation implies Cayley's theorem that any group can be embedded in a symmetric group (which is infinite when the group is). A finite group may act faithfully on a set of size much smaller than its cardinality (however such an action cannot be free). For instance the abelian 2-group ( Z / 2 Z ) n {\displaystyle (\mathbb {Z} /2\mathbb {Z} )^{n}} (of cardinality 2 n {\displaystyle 2^{n}} ) acts faithfully on a set of size 2 n {\displaystyle 2n} . This is not always the case, for example the cyclic group Z / 2 n Z {\displaystyle \mathbb {Z} /2^{n}\mathbb {Z} } cannot act faithfully on a set of size less than 2 n {\displaystyle 2^{n}} . In general, the smallest set on which a faithful action can be defined can vary greatly for groups of the same size. For example, three groups of size 120 are the symmetric group S 5 {\displaystyle S_{5}} , the icosahedral group A 5 × Z / 2 Z {\displaystyle A_{5}\times \mathbb {Z} /2\mathbb {Z} } and the cyclic group Z / 120 Z {\displaystyle \mathbb {Z} /120\mathbb {Z} } . The smallest sets on which faithful actions can be defined for these groups are of size 5, 7, and 16 respectively.
Transitivity properties The action of G {\displaystyle G} on X {\displaystyle X} is called transitive if for any two points x , y ∈ X {\displaystyle x,y\in X} there exists a g ∈ G {\displaystyle g\in G} so that g ⋅ x = y {\displaystyle g\cdot x=y} . The action is simply transitive (or sharply transitive, or regular) if it is both transitive and free. This means that given x , y ∈ X {\displaystyle x,y\in X} there is exactly one g ∈ G {\displaystyle g\in G} such that g ⋅ x = y {\displaystyle g\cdot x=y} . If X {\displaystyle X} is acted upon simply transitively by a group G {\displaystyle G} then it is called a principal homogeneous space for G {\displaystyle G} or a G {\displaystyle G} -torsor. For an integer n ≥ 1 {\displaystyle n\geq 1} , the action is n {\displaystyle n} -transitive if X {\displaystyle X} has at least n {\displaystyle n} elements, and for any pair of n {\displaystyle n} -tuples ( x 1 , … , x n ) , ( y 1 , … , y n ) ∈ X n {\displaystyle (x_{1},\ldots ,x_{n}),(y_{1},\ldots ,y_{n})\in X^{n}} with pairwise distinct entries (that is x i ≠ x j {\displaystyle x_{i}\neq x_{j}} , y i ≠ y j {\displaystyle y_{i}\neq y_{j}} when i ≠ j {\displaystyle i\neq j} ) there exists a g ∈ G {\displaystyle g\in G} such that g ⋅ x i = y i {\displaystyle g\cdot x_{i}=y_{i}} for i = 1 , … , n {\displaystyle i=1,\ldots ,n} . In other words, the action on the subset of X n {\displaystyle X^{n}} of tuples without repeated entries is transitive. For n = 2 , 3 {\displaystyle n=2,3} this is often called double, respectively triple, transitivity. The class of 2-transitive groups (that is, subgroups of a finite symmetric group whose action is 2-transitive) and more generally multiply transitive groups is well-studied in finite group theory. An action is sharply n {\displaystyle n} -transitive when the action on tuples without repeated entries in X n {\displaystyle X^{n}} is sharply transitive.
Examples The action of the symmetric group of X {\displaystyle X} is transitive, in fact n {\displaystyle n} -transitive for any n {\displaystyle n} up to the cardinality of X {\displaystyle X} . If X {\displaystyle X} has cardinality n {\displaystyle n} , the action of the alternating group is ( n − 2 ) {\displaystyle (n-2)} -transitive but not ( n − 1 ) {\displaystyle (n-1)} -transitive. The action of the general linear group of a vector space V {\displaystyle V} on the set V ∖ { 0 } {\displaystyle V\setminus \{0\}} of non-zero vectors is transitive, but not 2-transitive (similarly for the action of the special linear group if the dimension of V {\displaystyle V} is at least 2). The action of the orthogonal group of a Euclidean space on nonzero vectors is not transitive, but its action on the unit sphere is.
Homogeneity properties For an integer n ≥ 1 {\displaystyle n\geq 1} , the action of G {\displaystyle G} on X {\displaystyle X} is called n {\displaystyle n} -homogeneous if X {\displaystyle X} has at least n {\displaystyle n} elements, and for any pair of n {\displaystyle n} -subsets S , T ⊆ X {\displaystyle S,T\subseteq X} there exists a g ∈ G {\displaystyle g\in G} such that g S = T {\displaystyle gS=T} . In other words, the action on the set of n {\displaystyle n} -subsets of X {\displaystyle X} is transitive. An action is sharply n {\displaystyle n} -homogeneous when the action on n {\displaystyle n} -subsets of X {\displaystyle X} is sharply transitive. An n {\displaystyle n} -transitive action is also n {\displaystyle n} -homogeneous. As 1-subsets, 1-tuples and points are equivalent, the concepts of 1-homogeneity, 1-transitivity and transitivity coincide. A group that acts n {\displaystyle n} -homogeneously on a set X {\displaystyle X} of m {\displaystyle m} elements also acts
( m − n ) {\displaystyle (m-n)} -homogeneously, as g S = T {\displaystyle gS=T} implies g ( X ∖ S ) = X ∖ T {\displaystyle g\left(X\setminus S\right)=X\setminus T} .
Examples The action of the symmetric group of X {\displaystyle X} is n {\displaystyle n} -homogeneous for any n {\displaystyle n} up to the cardinality of X {\displaystyle X} . So is the action of the alternating group, which is ( n − 1 ) {\displaystyle (n-1)} -homogeneous (since it is transitive, and thus 1-homogeneous) without being ( n − 1 ) {\displaystyle (n-1)} -transitive. For n = 3 {\displaystyle n=3} , this has a geometric interpretation: the full group of Euclidean symmetries of an equilateral triangle acts as the full symmetric group on its vertices and hence 2-transitively thereon, while the subgroup of in-plane rotations (which correspond to alternating permutations) merely acts 2-homogeneously.
Primitive actions
The action of G {\displaystyle G} on X {\displaystyle X} is called primitive if there is no partition of X {\displaystyle X} preserved by all elements of G {\displaystyle G} apart from the trivial partitions (the partition in a single piece and its dual, the partition into singletons).
Topological properties Assume that X {\displaystyle X} is a topological space and the action of G {\displaystyle G} is by homeomorphisms. The action is wandering if every x ∈ X {\displaystyle x\in X} has a neighbourhood U {\displaystyle U} such that there are only finitely many g ∈ G {\displaystyle g\in G} with ( g ⋅ U ) ∩ U ≠ ∅ {\displaystyle (g\cdot U)\cap U\neq \emptyset } . More generally, a point x ∈ X {\displaystyle x\in X} is called a point of discontinuity for the action of G {\displaystyle G} if there is an open subset U ∋ x {\displaystyle U\ni x} such that there are only finitely many g ∈ G {\displaystyle g\in G} with ( g ⋅ U ) ∩ U ≠ ∅ {\displaystyle (g\cdot U)\cap U\neq \emptyset } . The domain of discontinuity of the action is the set of all points of discontinuity. Equivalently it is the largest G {\displaystyle G} -stable open subset Ω ⊂ X {\displaystyle \Omega \subset X} such that the action of G {\displaystyle G} on Ω {\displaystyle \Omega } is wandering. In a dynamical context this is also called a wandering set. The action is properly discontinuous if for every compact subset K ⊂ X {\displaystyle K\subset X} there are only finitely many g ∈ G {\displaystyle g\in G} such that ( g ⋅ K ) ∩ K ≠ ∅ {\displaystyle (g\cdot K)\cap K\neq \emptyset } . This is strictly stronger than wandering; for instance the action of Z {\displaystyle \mathbb {Z} } on R 2 ∖ { ( 0 , 0 ) } {\displaystyle \mathbb {R} ^{2}\backslash \{(0,0)\}} given by n ⋅ ( x , y ) = ( 2 n x , 2 − n y ) {\displaystyle n\cdot (x,y)=(2^{n}x,2^{-n}y)} is wandering and free but not properly discontinuous. The action by deck transformations of the fundamental group of a locally simply connected space on a universal cover is wandering and free. Such actions can be characterized by the following property: every x ∈ X {\displaystyle x\in X} has a neighbourhood U {\displaystyle U} such that ( g ⋅ U ) ∩ U = ∅ {\displaystyle (g\cdot U)\cap U=\emptyset } for every g ∈ G ∖ { e G } {\displaystyle g\in G\backslash \{e_{G}\}} . Actions with this property are sometimes called freely discontinuous, and the largest subset on which the action is freely discontinuous is then called the free regular set. An action of a group G {\displaystyle G} on a locally compact space X {\displaystyle X} is called cocompact if there exists a compact subset A ⊂ X {\displaystyle A\subset X} such that X = G ⋅ A {\displaystyle X=G\cdot A} . For a properly discontinuous action, cocompactness is equivalent to compactness of the quotient space X / G {\displaystyle X/G} .
Actions of topological groups
Now assume G {\displaystyle G} is a topological group and X {\displaystyle X} a topological space on which it acts by homeomorphisms. The action is said to be continuous if the map G × X → X {\displaystyle G\times X\rightarrow X} is continuous for the product topology. The action is said to be proper if the map G × X → X × X {\displaystyle G\times X\rightarrow X\times X} defined by ( g , x ) ↦ ( x , g ⋅ x ) {\displaystyle (g,x)\mapsto (x,g\cdot x)} is proper. This means that given compact sets K , K ′ {\displaystyle K,K'} the set of g ∈ G {\displaystyle g\in G} such that ( g ⋅ K ) ∩ K ′ ≠ ∅ {\displaystyle (g\cdot K)\cap K'\neq \varnothing } is compact. In particular, this is equivalent to proper discontinuity if G {\displaystyle G} is a discrete group. It is said to be locally free if there exists a neighbourhood U {\displaystyle U} of e G {\displaystyle e_{G}} such that g ⋅ x ≠ x {\displaystyle g\cdot x\neq x} for all x ∈ X {\displaystyle x\in X} and g ∈ U ∖ { e G } {\displaystyle g\in U\setminus \{e_{G}\}} . The action is said to be strongly continuous if the orbital map g ↦ g ⋅ x {\displaystyle g\mapsto g\cdot x} is continuous for every x ∈ X {\displaystyle x\in X} . Contrary to what the name suggests, this is a weaker property than continuity of the action. If G {\displaystyle G} is a Lie group and X {\displaystyle X} a differentiable manifold, then the subspace of smooth points for the action is the set of points x ∈ X {\displaystyle x\in X} such that the map g ↦ g ⋅ x {\displaystyle g\mapsto g\cdot x} is smooth. There is a well-developed theory of Lie group actions, i.e. action which are smooth on the whole space.
Linear actions
If g acts by linear transformations on a module over a commutative ring, the action is said to be irreducible if there are no proper nonzero g-invariant submodules. It is said to be semisimple if it decomposes as a direct sum of irreducible actions.
Orbits and stabilizers
Consider a group G acting on a set X. The orbit of an element x in X is the set of elements in X to which x can be moved by the elements of G. The orbit of x is denoted by G⋅x:
G ⋅ x = { g ⋅ x : g ∈ G } . {\displaystyle G{\cdot }x=\{g{\cdot }x:g\in G\}.}
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, that is, G⋅x = G⋅y. The group action is transitive if and only if it has exactly one orbit, that is, 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 (given that X is non-empty). The set of all orbits of X under the action of G is written as X / G (or, less frequently, as 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 XG, by contrast with the invariants (fixed points), denoted XG: 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 {\displaystyle Y} is a subset of X {\displaystyle X} , then G ⋅ Y {\displaystyle G\cdot Y} denotes the set { g ⋅ y ∣ g ∈ G , y ∈ Y } {\displaystyle \{g\cdot y\mid g\in G,y\in Y\}} . The subset Y {\displaystyle Y} is said to be invariant under G {\displaystyle G} if G ⋅ Y = Y {\displaystyle G\cdot Y=Y} (which is equivalent G ⋅ Y ⊆ Y {\displaystyle G\cdot Y\subseteq Y} ). In that case, G {\displaystyle G} also operates on Y {\displaystyle Y} by restricting the action to Y {\displaystyle Y} . The subset Y {\displaystyle Y} is called fixed under G {\displaystyle G} if g ⋅ y = y {\displaystyle g\cdot y=y} for all g {\displaystyle g} in G {\displaystyle G} and all y {\displaystyle y} in Y {\displaystyle Y} . Every subset that is fixed under G {\displaystyle G} is also invariant under G {\displaystyle G} , but not conversely. Every orbit is an invariant subset of X {\displaystyle X} on which G {\displaystyle G} acts transitively. Conversely, any invariant subset of X {\displaystyle X} is a union of orbits. The action of G {\displaystyle G} on X {\displaystyle X} is transitive if and only if all elements are equivalent, meaning that there is only one orbit. A G {\displaystyle G} -invariant element of X {\displaystyle X} is x ∈ X {\displaystyle x\in X} such that g ⋅ x = x {\displaystyle g\cdot x=x} for all g ∈ G {\displaystyle g\in G} . The set of all such x {\displaystyle x} is denoted X G {\displaystyle X^{G}} and called the G {\displaystyle G} -invariants of X {\displaystyle X} . When X {\displaystyle X} is a G {\displaystyle G} -module, X G {\displaystyle X^{G}} is the zeroth cohomology group of G {\displaystyle G} with coefficients in X {\displaystyle X} , and the higher cohomology groups are the derived functors of the functor of G {\displaystyle G} -invariants.
Fixed points and stabilizer subgroups Given g {\displaystyle g} in G {\displaystyle G} and x {\displaystyle x} in X {\displaystyle X} with g ⋅ x = x {\displaystyle g\cdot x=x} , it is said that " x {\displaystyle x} is a fixed point of g {\displaystyle g} " or that " g {\displaystyle g} fixes x {\displaystyle x} ". For every