Group action

If G is a group with identity element e, and X is a set, then a (left) group action α of G on X is a function

${\displaystyle \alpha \colon G\times X\to X,}$

that satisfies the following two axioms:[1]

Identity:	${\displaystyle \alpha (e,x)=x}$
Compatibility:	${\displaystyle \alpha \left(g,\alpha \left(h,x\right)\right)=\alpha \left(gh,x\right)}$

(with α(g, x) often shortened to gx or g ⋅ x when the action being considered is clear from context):

Identity:	${\displaystyle e\cdot x=x}$
Compatibility:	${\displaystyle g\cdot (h\cdot x)=(gh)\cdot x}$

for all g and h in G and all x in X.

The group G is said to act on X (from the left). A set X together with an action of G is called a (left) G-set.

From these two axioms, it follows that for any fixed g in G, the function from X to itself which maps x to g ⋅ x is a bijection, with inverse bijection the corresponding map for g⁻¹. Therefore, one may equivalently 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 itself.[2]

Likewise, a right group action of G on X is a function

${\displaystyle \alpha \colon X\times G\to X,}$

that satisfies the analogous axioms:[3]

Identity:	${\displaystyle \alpha (x,e)=x}$
Compatibility:	${\displaystyle \alpha \left(\alpha \left(x,g\right),h\right)=\alpha \left(x,gh\right)}$

(with α(x, g) often shortened to xg or x ⋅ g when the action being considered is clear from context)

Identity:	${\displaystyle x\cdot e=x}$
Compatibility:	${\displaystyle (x\cdot g)\cdot h=x\cdot (gh)}$

for all g and h in G and all x in X.

The difference between left and right actions is in the order in which a product gh acts on x. For a left action, h acts first, followed by g second. For a right action, g acts first, followed by h second. Because of the formula (gh)⁻¹ = h⁻¹g⁻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 on X can be considered as a left action of its opposite group G^op on X.

Thus, for establishing general properties of group actions, it suffices to consider only left actions. However, there are cases where this is not possible. For example, the multiplication of a group induces both a left action and a right action on the group itself—multiplication on the left and on the right, respectively.

The action of ${\displaystyle G}$ on ${\displaystyle X}$ is called transitive if for any two points ${\displaystyle x,y\in X}$ there exists a ${\displaystyle g\in G}$ so that ${\displaystyle g\cdot x=y}$.

The action is simply transitive (or sharply transitive, or regular) if it is both transitive and free. This means that given ${\displaystyle x,y\in X}$ the element ${\displaystyle g}$ in the definition of transitivity is unique. If ${\displaystyle X}$ is acted upon simply transitively by a group ${\displaystyle G}$ then it is called a principal homogeneous space for ${\displaystyle G}$ or a ${\displaystyle G}$-torsor.

For an integer ${\displaystyle n\geq 1}$, the action is n-transitive if ${\displaystyle X}$ has at least ${\displaystyle n}$ elements, and for any pair of ${\displaystyle n}$-tuples ${\displaystyle (x_{1},\ldots ,x_{n}),(y_{1},\ldots ,y_{n})\in X^{n}}$ with pairwise distinct entries (that is ${\displaystyle x_{i}\not =x_{j}}$, ${\displaystyle y_{i}\not =y_{j}}$ when ${\displaystyle i\not =j}$) there exists a ${\displaystyle g\in G}$ such that ${\displaystyle g\cdot x_{i}=y_{i}}$ for ${\displaystyle i=1,\ldots ,n}$. In other words the action on the subset of ${\displaystyle X^{n}}$ of tuples without repeated entries is transitive. For ${\displaystyle n=2,3}$ 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 ${\displaystyle n}$-transitive when the action on tuples without repeated entries in ${\displaystyle X^{n}}$ is sharply transitive.

The action of the symmetric group of ${\displaystyle X}$ is transitive, in fact ${\displaystyle n}$-transitive for any ${\displaystyle n}$ up to the cardinality of ${\displaystyle X}$. If ${\displaystyle X}$ has cardinality ${\displaystyle n,}$ the action of the alternating group is ${\displaystyle (n-2)}$-transitive but not ${\displaystyle (n-1)}$-transitive.

The action of the general linear group of a vector space ${\displaystyle V}$ on the set ${\displaystyle V\setminus \{0\}}$ of non-zero vectors is transitive, but not 2-transitive (similarly for the action of the special linear group if the dimension of ${\displaystyle v}$ is at least 2). The action of the orthogonal group of a Euclidean space is not transitive on nonzero vectors but it is on the unit sphere.

The action of ${\displaystyle G}$ on ${\displaystyle X}$ is called primitive if there is no partition of ${\displaystyle X}$ preserved by all elements of ${\displaystyle G}$ apart from the trivial partitions (the partition in a single piece and its dual, the partition into singletons).

Assume that ${\displaystyle X}$ is a topological space and the action of ${\displaystyle G}$ is by homeomorphisms.

The action is wandering if every ${\displaystyle x\in X}$ has a neighbourhood ${\displaystyle U}$ such that there are only finitely many ${\displaystyle g\in G}$ with ${\displaystyle g\cdot U\cap U\not =\emptyset }$.[4]

More generally, a point ${\displaystyle x\in X}$ is called a point of discontinuity for the action of ${\displaystyle G}$ if there is an open subset ${\displaystyle U\ni x}$ such that there are only finitely many ${\displaystyle g\in G}$ with ${\displaystyle g\cdot U\cap U\not =\emptyset }$. The domain of discontinuity of the action is the set of all points of discontinuity. Equivalently it is the largest ${\displaystyle G}$-stable open subset ${\displaystyle \Omega \subset X}$ such that the action of ${\displaystyle G}$ on ${\displaystyle \Omega }$ is wandering.[5] In a dynamical context this is also called wandering set.

The action is properly discontinuous if for every compact subset ${\displaystyle K\subset X}$ there are finitely many ${\displaystyle g\in G}$ such that ${\displaystyle g\cdot K\cap K\not =\emptyset }$. This is strictly stronger than wandering; for instance the action of ${\displaystyle \mathbb {Z} }$ on ${\displaystyle \mathbb {R} ^{2}\setminus \{(0,0)\}}$ given by ${\displaystyle n\cdot (x,y)=(2^{n}x,2^{-n}y)}$ is wandering and free but not properly discontinuous.[6]

The action by deck transformations of the fundamental group of a locally simply connected space on an covering space is wandering and free. Such actions can be characterized by the following property: every ${\displaystyle x\in X}$ has a neighbourhood ${\displaystyle U}$ such that ${\displaystyle g\cdot U\cap U=\emptyset }$ for every ${\displaystyle g\in G\setminus \{e_{G}\}}$.[7] 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.[8]

An action of a group ${\displaystyle G}$ on a locally compact space ${\displaystyle X}$ is called cocompact if there exists a compact subset ${\displaystyle A\subset X}$ such that ${\displaystyle X=G\cdot A}$. For a properly discontinuous action, cocompactness is equivalent to compactness of the quotient space ${\displaystyle G\backslash X}$.

Now assume ${\displaystyle G}$ is a topological group and ${\displaystyle X}$ a topological space on which it acts by homeomorphisms. The action is said to be continuous if the map ${\displaystyle G\times X\to X}$ is continuous for the product topology.

The action is said to be proper if the map ${\displaystyle G\times X\to X\times X}$ defined by ${\displaystyle (g,x)\mapsto (x,g\cdot x)}$ is proper.[9] This means that given compact sets ${\displaystyle K,K'}$ the set of ${\displaystyle g\in G}$ such that ${\displaystyle g\cdot K\cap K'\not =\emptyset }$ is compact. In particular, this is equivalent to proper discontinuity when ${\displaystyle G}$ is a discrete group.

It is said to be locally free if there exists a neighbourhood ${\displaystyle U}$ of ${\displaystyle e_{G}}$ such that ${\displaystyle g\cdot x\not =x}$ for all ${\displaystyle x\in X}$ and ${\displaystyle g\in U\setminus \{e_{G}\}}$.

The action is said to be strongly continuous if the orbital map ${\displaystyle g\mapsto g\cdot x}$ is continuous for every ${\displaystyle x\in X}$. Contrary to what the name suggests, this is a weaker property than continuity of the action.[10]

If ${\displaystyle G}$ is a Lie group and ${\displaystyle X}$ a differentiable manifold, then the subspace of smooth points for the action is the set of points ${\displaystyle x\in X}$ such that the map ${\displaystyle x\mapsto g\cdot x}$ is smooth.

If ${\displaystyle g}$ acts by linear transformations on a module over a commutative ring, the action is said to be irréducible if there are no proper nonzero ${\displaystyle g}$-invariant submodules. It is said to be semisimple if it decomposes as a direct sum of irreducible actions.

If Y is a subset of X, then ${\displaystyle G\cdot Y}$ denotes the set ${\displaystyle \{g\cdot y:g\in G{\text{ and }}y\in Y\}.}$ The subset Y is said to be invariant under G if ${\displaystyle G\cdot Y=Y}$ (which is equivalent to ${\displaystyle G\cdot Y\subseteq Y}$). In that case, G also operates on Y by restricting the action to Y. The subset Y is called fixed under G if ${\displaystyle g\cdot y=y}$ for all g in G and all y in Y. Every subset that is fixed under G is also invariant under G, but not conversely.

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

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

Given g in G and x in X with ${\displaystyle g\cdot x=x,}$ it is said that "x is a fixed point of g" or that "g fixes x". For every x in X, the stabilizer subgroup of G with respect to x (also called the isotropy group or little group[11]) is the set of all elements in G that fix x:

$${\displaystyle G_{x}=\{g\in G:g\cdot x=x\}.}$$

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 with the symmetric group, ${\displaystyle G\to \operatorname {Sym} (X),}$ is given by the intersection of the stabilizers G_x 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 ${\displaystyle g}$ be a group element such that ${\displaystyle y=g\cdot x.}$ Then the two stabilizer groups ${\displaystyle G_{x}}$ and ${\displaystyle G_{y}}$ are related by ${\displaystyle G_{y}=gG_{x}g^{-1}.}$ Proof: by definition, ${\displaystyle h\in G_{y}}$ if and only if ${\displaystyle h\cdot (g\cdot x)=g\cdot x.}$ Applying ${\displaystyle g^{-1}}$ to both sides of this equality yields ${\displaystyle \left(g^{-1}hg\right)\cdot x=x;}$ that is, ${\displaystyle g^{-1}hg\in G_{x}.}$ An opposite inclusion follows similarly by taking ${\displaystyle h\in G_{x}}$ and supposing ${\displaystyle x=g^{-1}\cdot y.}$

The above says that the stabilizers of elements in the same orbit are conjugate to each other. Thus, to each orbit, we can associate a conjugacy class of a subgroup of G (that is, the set of all conjugates of the subgroup). Let ${\displaystyle (H)}$ denote the conjugacy class of H. Then the orbit O has type ${\displaystyle (H)}$ if the stabilizer ${\displaystyle G_{x}}$ of some/any x in O belongs to ${\displaystyle (H)}$. A maximal orbit type is often called a principal orbit type.

Orbits and stabilizers are closely related. For a fixed x in X, consider the map ${\displaystyle f:G\to X}$ given by ${\displaystyle g\mapsto g\cdot x.}$ By definition the image ${\displaystyle f(G)}$ of this map is the orbit ${\displaystyle G\cdot x.}$ The condition for two elements to have the same image is

$${\displaystyle f(g)=f(h)\iff g\cdot x=h\cdot x\iff g^{-1}h\cdot x=x\iff g^{-1}h\in G_{x}\iff h\in gG_{x}.}$$

In other words, ${\displaystyle f(g)=f(h)}$ if and only if ${\displaystyle g}$ and ${\displaystyle h}$ lie in the same coset for the stabilizer subgroup ${\displaystyle G_{x}}$. Thus, the fiber ${\displaystyle f^{-1}(\{y\})}$ of f over any y in G·x is contained in such a coset, and every such coset also occurs as a fiber. Therefore f induces a bijection between the set ${\displaystyle G/G_{x}}$ of cosets for the stabilizer subgroup and the orbit ${\displaystyle G\cdot x,}$ which sends ${\displaystyle gG_{x}\mapsto g\cdot x}$.[12] This result is known as the orbit-stabilizer theorem.

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

$${\displaystyle |G\cdot x|=[G\,:\,G_{x}]=|G|/|G_{x}|,}$$

in other words the length of the orbit of x times the order of its stabilizer is the order of the group. In particular that implies that the orbit length is a divisor of the group order.

Example: Let G be a group of prime order p acting on a set X with k elements. Since each orbit has either 1 or p elements, there are at least ${\displaystyle k{\bmod {p}}}$ orbits of length 1 which are G-invariant elements.

This result is especially useful since it can be employed for counting arguments (typically in situations where X is finite as well).

Cubical graph with vertices labeled

Example: We can use the orbit-stabilizer theorem to count the automorphisms of a graph. Consider the cubical graph as pictured, and let G denote its automorphism group. Then G acts on the set of vertices {1, 2, ..., 8}, and this action is transitive as can be seen by composing rotations about the center of the cube. Thus, by the orbit-stabilizer theorem, ${\displaystyle |G|=|G\cdot 1||G_{1}|=8|G_{1}|.}$ Applying the theorem now to the stabilizer ${\displaystyle G_{1},}$ we can obtain ${\displaystyle |G_{1}|=|(G_{1})\cdot 2||(G_{1})_{2}|.}$ Any element of G that fixes 1 must send 2 to either 2, 4, or 5. As an example of such automorphisms consider the rotation around the diagonal axis through 1 and 7 by ${\displaystyle 2\pi /3}$ which permutes 2,4,5 and 3,6,8, and fixes 1 and 7. Thus, ${\displaystyle \left|(G_{1})\cdot 2\right|=3.}$ Applying the theorem a third time gives ${\displaystyle |\left(G_{1}\right)_{2}|=|\left(\left(G_{1}\right)_{2}\right)\cdot 3||\left(\left(G_{1}\right)_{2}\right)_{3}|.}$ Any element of G that fixes 1 and 2 must send 3 to either 3 or 6. Reflecting the cube at the plane through 1,2,7 and 8 is such an automorphism sending 3 to 6, thus ${\displaystyle \left|\left(\left(G_{1}\right)_{2}\right)\cdot 3\right|=2}$. One also sees that ${\displaystyle \left(\left(G_{1}\right)_{2}\right)_{3}}$ consists only of the identity automorphism, as any element of G fixing 1, 2 and 3 must also fix all other vertices, since they are determined by their adjacency to 1, 2 and 3. Combining the preceding calculations, we can now obtain ${\displaystyle |G|=8\cdot 3\cdot 2\cdot 1=48.}$

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

$${\displaystyle |X/G|={\frac {1}{|G|}}\sum _{g\in G}|X^{g}|,}$$

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.

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.