Group action · aiwiki.page

A group action is a rule assigning transformations of a set to the elements of a group, with the identity acting trivially and multiplication corresponding to composition. It connects the abstract operations studied in group theory with concrete symmetries, such as permutations of objects, rotations of figures, and transformations of vectors. The same group can act on many different sets, so an action specifies not only a group but also how its elements transform particular objects. (courses.maths.ox.ac.uk)

Definition and conventions

Let $G$ be a group with identity element $e$ , and let $X$ be a set. A left action of $G$ on $X$ is a function

G\times X\longrightarrow X,\qquad (g,x)\longmapsto g\cdot x,

satisfying

e\cdot x=x,\qquad (gh)\cdot x=g\cdot(h\cdot x)

for all $g,h\in G$ and $x\in X$ . The second condition means that $h$ acts first and $g$ acts second. A set equipped with an action of $G$ is called a $G$ -set. (jmilne.org)

Every transformation $x\mapsto g\cdot x$ is a bijection, with inverse $x\mapsto g^{-1}\cdot x$ . Equivalently, an action is a group homomorphism

\rho:G\longrightarrow \operatorname{Sym}(X),

where $\operatorname{Sym}(X)$ is the symmetric group of all permutations of $X$ , with multiplication given by composition. Thus $\rho(gh)=\rho(g)\circ\rho(h)$ . (courses.maths.ox.ac.uk)

A right action, written $x\cdot g$ , instead satisfies

x\cdot e=x,\qquad (x\cdot g)\cdot h=x\cdot(gh).

A left action gives a right action through $x\cdot g=g^{-1}\cdot x$ . The inverse is essential: simply moving the group symbol to the other side does not generally preserve the action law. (jmilne.org)

Orbits, stabilizers, and kernels

The orbit of $x\in X$ is

G\cdot x=\{g\cdot x:g\in G\}.

It consists of all points reachable from $x$ . Being in the same orbit is an equivalence relation, so the orbits partition $X$ . Their collection is the orbit set, written $X/G$ , and is a particular quotient set. (ocw.mit.edu)

The stabilizer, or isotropy subgroup, of $x$ is

G_x=\{g\in G:g\cdot x=x\}.

It is a subgroup of $G$ . Stabilizers of points in the same orbit are conjugate:

G_{g\cdot x}=gG_xg^{-1}.

The kernel of the action consists of the elements fixing every point:

\ker\rho=\bigcap_{x\in X}G_x.

It is a normal subgroup of $G$ . Unlike a point stabilizer, it measures which group elements are invisible throughout the entire action. (courses.maths.ox.ac.uk)

Important classes of actions include:

Transitive: $X$ is nonempty and has one orbit.
Faithful, or effective: $\ker\rho=\{e\}$ .
Free: every point stabilizer is $\{e\}$ .
Regular, or simply transitive: the action is both free and transitive.

A free action on a nonempty set is faithful, but a faithful action need not be free. (jmilne.org)

For example, the symmetries of a square act faithfully and transitively on its four vertices. The action is not free: reflection in a diagonal fixes the two vertices on that diagonal. Restricting the action to the four rotations makes it regular. These properties follow directly by checking the images of the vertices under each symmetry. (ocw.mit.edu)

The orbit–stabilizer theorem

For any $x\in X$ , there is a natural bijection

G/G_x\longrightarrow G\cdot x,\qquad gG_x\longmapsto g\cdot x,

where $G/G_x$ denotes the set of left cosets, not necessarily a quotient group. Consequently,

|G\cdot x|=[G:G_x].

For finite $G$ , the orbit–stabilizer theorem becomes

|G|=|G\cdot x|\,|G_x|.

Thus orbit sizes divide the order of a finite group. (ocw.mit.edu)

To see why the coset correspondence works, observe that

g\cdot x=h\cdot x \iff h^{-1}g\in G_x \iff gG_x=hG_x.

The map therefore identifies exactly those group elements giving the same image of $x$ , and every point in the orbit has such an image. (ocw.mit.edu)

More generally, every transitive $G$ -set is equivalent to the action on $G/H$ by left multiplication for some subgroup $H$ . Two such transitive actions are isomorphic precisely when their stabilizer subgroups are conjugate. An arbitrary $G$ -set is a disjoint union of these transitive pieces. (jmilne.org)

Fundamental examples

Permutations. The group $S_n$ acts on $\{1,\ldots,n\}$ by evaluating permutations. It can also act on subsets, ordered tuples, or arrangements of labels. Choosing a different underlying set produces a different action of the same group. (courses.maths.ox.ac.uk)

Linear transformations. Invertible linear maps of a vector space $V$ form the general linear group $\operatorname{GL}(V)$ , which acts on $V$ by evaluation. A homomorphism $G\to\operatorname{GL}(V)$ is a linear representation, the central object of representation theory. (jmilne.org)

Multiplication and conjugation. Every group acts on itself by left multiplication, $g\cdot x=gx$ . This action is regular and embeds the group into a permutation group, as expressed by Cayley’s theorem. Another action is conjugation,

g\cdot x=gxg^{-1}.

Its orbits are conjugacy classes, and its stabilizers are centralizers. (courses.maths.ox.ac.uk)

Counting objects up to symmetry

For a finite group acting on a finite set, Burnside’s lemma gives

|X/G|=\frac{1}{|G|}\sum_{g\in G}|\operatorname{Fix}(g)|, \qquad \operatorname{Fix}(g)=\{x:g\cdot x=x\}.

It counts distinct objects up to the chosen symmetries by averaging fixed-point counts. This is a basic tool in combinatorics. (courses.maths.ox.ac.uk)

For example, color the four vertices of a square using two colors, allowing repetitions, and identify colorings related by rotation. There are $16$ labeled colorings. The identity fixes all $16$ ; each quarter-turn fixes $2$ ; and the half-turn fixes $4$ . Substitution gives

\frac{16+2+4+2}{4}=6

rotation classes. Reflections are excluded here; including them changes the acting group and requires another fixed-point calculation. This example illustrates why dividing the number of objects by the group order is generally insufficient: different objects can have different stabilizers. (courses.maths.ox.ac.uk)

Actions preserving additional structure

When $X$ carries additional structure, one generally requires the action to preserve it. For a topological space and a topological group, a continuous action requires the joint map $G\times X\to X$ to be continuous. For a Lie group acting on a smooth manifold, a smooth action requires that map to be smooth. (math.toronto.edu)

The orbit set then carries the quotient topology, but it need not remain a manifold. A central sufficient condition is a smooth, free, proper action. Here properness means that

G\times M\longrightarrow M\times M,\qquad (g,x)\longmapsto(g\cdot x,x)

is a proper map, meaning inverse images of compact sets are compact. Under these hypotheses, $M/G$ has a natural smooth manifold structure, and the projection $M\to M/G$ is a principal bundle. Freeness alone does not guarantee such a quotient. (math.toronto.edu)

Maps between spaces with actions can also respect symmetry. A map $f:X\to Y$ is equivariant when

f(g\cdot x)=g\cdot f(x).

A bijective equivariant map is an isomorphism of $G$ -sets. Equivariance distinguishes a correspondence that preserves the specified action from an arbitrary correspondence between the underlying sets. (jmilne.org)

References

Group Theoryjmilne.org
RES.18-011 (Fall 2021) Full Lecture Notes: Algebra I Student Notesocw.mit.edu
M1: Groups and Group Actions (2021-22)courses.maths.ox.ac.uk
Groups and Group Actions: Lecture Notescourses.maths.ox.ac.uk
Group Actionsmath.toronto.edu