Group action

In mathematics, 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 acts also on triangles by transforming triangles into triangles.

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

Formally, a group action of a group $G$ on a set $S$ is a group homomorphism from $G$ to some group (under function composition) of functions from $S$ to itself.

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 $GL(n, K)$ , the group of the invertible matrices of dimension $n$ over a field $K$ .

The symmetric group $S n$ acts on any set with $n$ 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[edit]

Left group action[edit]

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

The trivial action of any group $G$ on any set $X$ is defined by $g \cdot x = x$ for all $g$ in $G$ and all $x$ in $X$ ; that is, every group element induces the on $X$ .^[12]

identity permutation

In every group $G$ , left multiplication is an action of $G$ on $G$ : $g \cdot x = gx$ for all $g$ , $x$ in $G$ . This action is free and transitive (regular), and forms the basis of a rapid proof of – that every group is isomorphic to a subgroup of the symmetric group of permutations of the set $G$ .

Cayley's theorem

In every group $G$ with subgroup $H$ , left multiplication is an action of $G$ on the set of cosets $G / H$ : $g \cdot aH = gaH$ for all $g$ , $a$ in $G$ . In particular if $H$ contains no nontrivial of $G$ this induces an isomorphism from $G$ to a subgroup of the permutation group of degree $[G : H]$ .

normal subgroups

In every group $G$ , is an action of $G$ on $G$ : $g \cdot x = gxg -1$ . An exponential notation is commonly used for the right-action variant: $x g = g -1 xg$ ; it satisfies ( $x g) h = x gh$ .

conjugation

In every group $G$ with subgroup $H$ , conjugation is an action of $G$ on conjugates of $H$ : $g \cdot K = gKg -1$ for all $g$ in $G$ and $K$ conjugates of $H$ .

An action of $Z$ on a set $X$ uniquely determines and is determined by an of $X$ , given by the action of 1. Similarly, an action of $Z / 2 Z$ on $X$ is equivalent to the data of an involution of $X$ .

automorphism

The symmetric group $S n$ and its subgroups act on the set ${1, ..., n}$ by permuting its elements

The 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.

symmetry group

The symmetry group of any geometrical object acts on the set of points of that object.

For a $V$ over a field $F$ with group of units $F *$ , the mapping $F * \times V \to V$ given by $a \times (x 1, x 2, ..., x n) \mapsto (ax 1, ax 2, ..., ax n)$ is a group action called scalar multiplication.

coordinate space

The automorphism group of a vector space (or , or group, or ring ...) acts on the vector space (or set of vertices of the graph, or group, or ring ...).

graph

The general linear group $GL(n, K)$ and its subgroups, particularly its (including the special linear group $SL(n, K)$ , orthogonal group $O(n, K)$ , special orthogonal group $SO(n, K)$ , and symplectic group $Sp(n, K)$ ) are Lie groups that act on the vector space $K n$ . The group operations are given by multiplying the matrices from the groups with the vectors from $K n$ .

Lie subgroups

The general linear group $GL(n, Z)$ acts on $Z n$ by natural matrix action. The orbits of its action are classified by the of coordinates of the vector in $Z n$ .

greatest common divisor

The acts transitively on the points of an affine space, and the subgroup V of the affine group (that is, a vector space) has transitive and free (that is, regular) action on these points;^[13] indeed this can be used to give a definition of an affine space.

affine group

The $PGL(n + 1, K)$ and its subgroups, particularly its Lie subgroups, which are Lie groups that act on the projective space $P n (K)$ . This is a quotient of the action of the general linear group on projective space. Particularly notable is $PGL(2, K)$ , the symmetries of the projective line, which is sharply 3-transitive, preserving the cross ratio; the Möbius group $PGL(2, C)$ is of particular interest.

projective linear group

The of the plane act on the set of 2D images and patterns, such as wallpaper patterns. The definition can be made more precise by specifying what is meant by image or pattern, for example, a function of position with values in a set of colors. Isometries are in fact one example of affine group (action).

isometries

The sets acted on by a group $G$ comprise the of $G$ -sets in which the objects are $G$ -sets and the morphisms are $G$ -set homomorphisms: functions $f : X \to Y$ such that $g \cdot(f (x)) = f (g \cdot x)$ for every $g$ in $G$ .

Galois group

The additive group of the $(R, +)$ acts on the phase space of "well-behaved" systems in classical mechanics (and in more general dynamical systems) by time translation: 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.

real numbers

The additive group of the real numbers $(R, +)$ acts on the set of real functions of a real variable in various ways, with $(t \cdot f)(x)$ equal to, for example, $f (x + t)$ , $f (x) + t$ , $f (xe t)$ , $f (x) e t$ , $f (x + t) e t$ , or $f (xe t) + t$ , but not $f (xe t + t)$ .

Given a group action of $G$ on $X$ , we can define an induced action of $G$ on the of $X$ , by setting $g \cdot U = {g \cdot u : u \in U}$ 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.

power set

The with norm 1 (the versors), as a multiplicative group, act on $R 3$ : for any such quaternion $z = cos α /2 + v sin α /2$ , the mapping $f (x) = z x z *$ is a counterclockwise rotation through an angle $α$ about an axis given by a unit vector $v$ ; $z$ is the same rotation; see quaternions and spatial rotation. This is not a faithful action because the quaternion $-1$ leaves all points where they were, as does the quaternion $1$ .

quaternions

Given left $G$ -sets $X$ , $Y$ , there is a left $G$ -set $Y X$ whose elements are $G$ -equivariant maps $α : X \times G \to Y$ , and with left $G$ -action given by $g \cdot α = α \circ (id X \times - g)$ (where " $- g$ " indicates right multiplication by $g$ ). This $G$ -set has the property that its fixed points correspond to equivariant maps $X \to Y$ ; more generally, it is an in the category of $G$ -sets.

exponential object

Every regular $G$ action is isomorphic to the action of $G$ on $G$ given by left multiplication.

Every free $G$ action is isomorphic to $G \times S$ , where $S$ is some set and $G$ acts on $G \times S$ by left multiplication on the first coordinate. ( $S$ can be taken to be the set of orbits $X / G$ .)

Every transitive $G$ action is isomorphic to left multiplication by $G$ on the set of left cosets of some subgroup $H$ of $G$ . ( $H$ can be taken to be the stabilizer group of any element of the original $G$ -set.)

If $X$ and $Y$ are two $G$ -sets, a morphism from $X$ to $Y$ is a function $f : X \to Y$ such that $f (g \cdot x) = g \cdot f (x)$ 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. If a morphism $f$ is bijective, then its inverse is also a morphism. In this case $f$ is called an isomorphism, and the two $G$ -sets $X$ and $Y$ are called isomorphic; for all practical purposes, isomorphic $G$ -sets are indistinguishable.

Some example isomorphisms:

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).

Variants and generalizations[edit]

We 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, we 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.

We can view a group $G$ as a category with a single object in which every morphism is invertible.^[14] A (left) group action is then nothing but a (covariant) functor from $G$ to the category of sets, and a group representation is a functor from $G$ to the category of vector spaces.^[15] A morphism between $G$ -sets is then a natural transformation between the group action functors.^[16] In analogy, an action of a groupoid is a functor from the groupoid to the category of sets or to some other category.

In addition to continuous actions of topological groups on topological spaces, one also often considers smooth actions of Lie groups on smooth manifolds, regular actions of algebraic groups on algebraic varieties, and actions of group schemes on schemes. All of these are examples of group objects acting on objects of their respective category.

Orbit of a fundamental spherical triangle (marked in red) under action of the full octahedral group.

Orbit of a fundamental spherical triangle (marked in red) under action of the full icosahedral group.

Gain graph

Group with operators

Measurable group action

Monoid action

Young–Deruyts development

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Group action

Definition[edit]

Left group action[edit]

identity permutation

Cayley's theorem

normal subgroups

conjugation

automorphism

symmetry group

coordinate space

graph

Lie subgroups

greatest common divisor

affine group

projective linear group

isometries

category

Galois group

real numbers

power set

quaternions

exponential object

Variants and generalizations[edit]

Gain graph

Group with operators

Measurable group action

Monoid action

Young–Deruyts development

Aschbacher, Michael

ISBN

ISBN

Hatcher, Allen

ISBN

ISBN

ISBN

Zbl

doi

The geometry and topology of three-manifolds

Zbl

Transformation groups

"Action of a group on a manifold"

Weisstein, Eric W.