Group action

Inmathematics,many sets oftransformationsform agroupunderfunction composition;for example, therotationsaround a point in the plane. It is often useful to consider the group as anabstract group,and to say that one has agroup actionof 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 astructureacts also on various related structures; for example, the above rotation groupactsalso on triangles by transforming triangles into triangles.

Formally, agroup actionof a group $G$ on aset $S$ is agroup homomorphismfrom $G$ to some group (underfunction 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 ofEuclidean isometriesacts onEuclidean spaceand also on the figures drawn in it; in particular, it acts on the set of alltriangles.Similarly, the group ofsymmetriesof apolyhedronacts on thevertices,theedges,and thefacesof the polyhedron.

A group action on avector spaceis called arepresentationof the group. In the case of a finite-dimensional vector space, it allows one to identify many groups withsubgroupsof thegeneral linear group $GL(n, K)$ ,the group of theinvertible matricesofdimension $n$ over afield $K$ .

Thesymmetric group $S n$ acts on anysetwith $n$ elements by permuting the elements of the set. Although the group of allpermutationsof 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 samecardinality.

Definition[edit]

Left group action[edit]

If $G$ is agroupwithidentity element $e$ ,and $X$ is a set, then a (left)group action $α$ of $G$ on $X$ is afunction

\ Alpha \colon G\times X\to X,

that satisfies the following twoaxioms:^[1]

Identity:	$\ Alpha (e,x)=x$
Compatibility:	$\ Alpha (g,\ Alpha (h,x))=\ Alpha (gh,x)$

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

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

It can be notationally convenient tocurrythe action $α$ ,so that, instead, one has a collection oftransformations $α g : X \to X$ ,with one transformation $α g$ for each group element $g \in G$ .The identity and compatibility relations then read

\ Alpha _{e}(x)=x

and

\ Alpha _{g}(\ Alpha _{h}(x))=(\ Alpha _{g}\circ \ Alpha _{h})(x)=\ Alpha _{gh}(x)

with $\circ$ beingfunction composition.The second axiom then states that the function composition is compatible with the group multiplication; they form acommutative diagram.This axiom can be shortened even further, and written as $α g \circ α h = α gh$ .

With the above understanding, it is very common to avoid writing $α$ entirely, and to replace it with either a dot, or with nothing at all. Thus, $α (g, x)$ can be shortened to $g \cdot x$ or $gx$ ,especially when the action is clear from context. The axioms are then

e{\cdot }x=x

g{\cdot }(h{\cdot }x)=(gh){\cdot }x

From these two axioms, it follows that for any fixed $g$ in $G$ ,the function from $X$ to itself which maps $x$ to $g \cdot x$ is abijection,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]

Right group action[edit]

Likewise, aright group actionof $G$ on $X$ is a function

\ Alpha \colon X\times G\to X,

that satisfies the analogous axioms:^[3]

Identity:	$\ Alpha (x,e)=x$
Compatibility:	$\ Alpha (\ Alpha (x,g),h)=\ Alpha (x,gh)$

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

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

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 $(gh) -1 = h -1 g -1$ ,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 $G$ on $X$ can be considered as a left action of itsopposite 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 groupinducesboth a left action and a right action on the group itself—multiplication on the left and on the right, respectively.

Notable properties of actions[edit]

Let $G$ be a group acting on a set $X$ .The action is calledfaithfuloreffectiveif $g \cdot x = x$ for all $x \in X$ implies that $g = e G$ .Equivalently, thehomomorphismfrom $G$ to the group of bijections of $X$ corresponding to the action isinjective.

The action is calledfree(orsemiregularorfixed-point free) if the statement that $g \cdot x = x$ for some $x \in X$ already implies that $g = e G$ .In other words, no non-trivial element of $G$ fixes a point of $X$ .This is a much stronger property than faithfulness.

For example, the action of any group on itself by left multiplication is free. This observation impliesCayley's theoremthat any group can beembeddedin 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 $(Z / 2 Z) n$ (of cardinality $2 n$ ) acts faithfully on a set of size $2 n$ .This is not always the case, for example thecyclic group $Z / 2 n Z$ cannot act faithfully on a set of size less than $2 n$ .

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 $S 5$ ,the icosahedral group $A 5 \times Z / 2 Z$ and the cyclic group $Z / 120 Z$ .The smallest sets on which faithful actions can be defined for these groups are of size 5, 7, and 16 respectively.

Transitivity properties[edit]

The action of $G$ on $X$ is calledtransitiveif for any two points $x, y \in X$ there exists a $g \in G$ so that $g \cdot x = y$ .

The action issimply transitive(orsharply transitive,orregular) if it is both transitive and free. This means that given $x, y \in X$ the element $g$ in the definition of transitivity is unique. If $X$ is acted upon simply transitively by a group $G$ then it is called aprincipal homogeneous spacefor $G$ or a $G$ -torsor.

For an integer $n \geq 1$ ,the action is $n$ -transitiveif $X$ has at least $n$ elements, and for any pair of $n$ -tuples $(x 1,..., x n), (y 1,..., y n) \in X n$ with pairwise distinct entries (that is $x i \neq x j$ , $y i \neq y j$ when $i \neq j$ ) there exists a $g \in G$ such that $g \cdot x i = y i$ for $i = 1,..., n$ .In other words the action on the subset of $X n$ of tuples without repeated entries is transitive. For $n = 2, 3$ this is often called double, respectively triple, transitivity. The class of2-transitive groups(that is, subgroups of a finite symmetric group whose action is 2-transitive) and more generallymultiply transitive groupsis well-studied in finite group theory.

An action issharply $n$ -transitivewhen the action on tuples without repeated entries in $X n$ is sharply transitive.

Examples[edit]

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

The action of thegeneral linear groupof a vector space $V$ on the set $V ∖ {0}$ of non-zero vectors is transitive, but not 2-transitive (similarly for the action of thespecial linear groupif the dimension of $v$ is at least 2). The action of theorthogonal groupof a Euclidean space is not transitive on nonzero vectors but it is on theunit sphere.

Primitive actions[edit]

The action of $G$ on $X$ is calledprimitiveif there is nopartitionof $X$ preserved by all elements of $G$ apart from the trivial partitions (the partition in a single piece and itsdual,the partition intosingletons).

Topological properties[edit]

Assume that $X$ is atopological spaceand the action of $G$ is byhomeomorphisms.

The action iswanderingif every $x \in X$ has aneighbourhood $U$ such that there are only finitely many $g \in G$ with $g \cdot U \cap U \neq \emptyset$ .^[4]

More generally, a point $x \in X$ is called a point of discontinuity for the action of $G$ if there is an open subset $U ∋ x$ such that there are only finitely many $g \in G$ with $g \cdot U \cap U \neq \emptyset$ .Thedomain of discontinuityof the action is the set of all points of discontinuity. Equivalently it is the largest $G$ -stable open subset $Ω \subset X$ such that the action of $G$ on $Ω$ is wandering.^[5]In a dynamical context this is also called awandering set.

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

The action bydeck transformationsof thefundamental groupof a locally simply connected space on ancovering spaceis wandering and free. Such actions can be characterized by the following property: every $x \in X$ has a neighbourhood $U$ such that $g \cdot U \cap U = \emptyset$ for every $g \in G ∖ {e G}$ .^[7]Actions with this property are sometimes calledfreely discontinuous,and the largest subset on which the action is freely discontinuous is then called thefree regular set.^[8]

An action of a group $G$ on alocally compact space $X$ is calledcocompactif there exists a compact subset $A \subset X$ such that $X = G \cdot A$ .For a properly discontinuous action, cocompactness is equivalent to compactness of thequotient space $G\X$ .

Actions of topological groups[edit]

Now assume $G$ is atopological groupand $X$ a topological space on which it acts by homeomorphisms. The action is said to becontinuousif the map $G \times X \to X$ is continuous for theproduct topology.

The action is said to beproperif the map $G \times X \to X \times X$ defined by $(g, x) \mapsto (x, g \cdot x)$ isproper.^[9]This means that given compact sets $K, K'$ the set of $g \in G$ such that $g \cdot K \cap K' \neq \emptyset$ is compact. In particular, this is equivalent to proper discontinuity $G$ is adiscrete group.

It is said to belocally freeif there exists a neighbourhood $U$ of $e G$ such that $g \cdot x \neq x$ for all $x \in X$ and $g \in U ∖ {e G}$ .

The action is said to bestrongly continuousif the orbital map $g \mapsto g \cdot x$ is continuous for every $x \in X$ .Contrary to what the name suggests, this is a weaker property than continuity of the action.^{[citation needed]}

If $G$ is aLie groupand $X$ adifferentiable manifold,then the subspace ofsmooth pointsfor the action is the set of points $x \in X$ such that the map $g \mapsto g \cdot x$ issmooth.There is a well-developed theory ofLie group actions,i.e. action which are smooth on the whole space.

Linear actions[edit]

If $g$ acts by linear transformations on amoduleover acommutative ring,the action is said to beirreducibleif there are no proper nonzero $g$ -invariant submodules. It is said to besemisimpleif it decomposes as adirect sumof irreducible actions.

Orbits and stabilizers[edit]

Consider a group $G$ acting on a set $X$ .Theorbitof 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 $G \cdot x$ : $G{\cdot }x=\{g{\cdot }x:g\in G\}.$

The defining properties of a group guarantee that the set of orbits of (points $x$ in) $X$ under the action of $G$ form apartitionof $X$ .The associatedequivalence relationis defined by saying $x ~ y$ if and only ifthere exists a $g$ in $G$ with $g \cdot x = y$ .The orbits are then theequivalence classesunder this relation; two elements $x$ and $y$ are equivalent if and only if their orbits are the same, that is, $G \cdot x = G \cdot y$ .

The group action istransitiveif and only if it has exactly one orbit, that is, if there exists $x$ in $X$ with $G \cdot x = X$ .This is the case if and only if $G \cdot x = X$ forall $x$ 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 thequotientof the action. In geometric situations it may be called theorbit space,while in algebraic situations it may be called the space ofcoinvariants,and written $X G$ ,by contrast with the invariants (fixed points), denoted $X G$ :the coinvariants are aquotientwhile the invariants are asubset.The coinvariant terminology and notation are used particularly ingroup cohomologyandgroup homology,which use the same superscript/subscript convention.

Invariant subsets[edit]

If $Y$ is asubsetof $X$ ,then $G \cdot Y$ denotes the set ${g \cdot y : g \in G and y \in Y}$ .The subset $Y$ is said to beinvariant under $G$ if $G \cdot Y = Y$ (which is equivalent $G \cdot Y \subseteq Y$ ). In that case, $G$ also operates on $Y$ byrestrictingthe action to $Y$ .The subset $Y$ is calledfixed under $G$ if $g \cdot y = y$ 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 $G$ actstransitively.Conversely, any invariant subset of $X$ is a union of orbits. The action of $G$ on $X$ istransitiveif and only if all elements are equivalent, meaning that there is only one orbit.

A $G$ -invariantelement of $X$ is $x \in X$ such that $g \cdot x = x$ for all $g \in G$ .The set of all such $x$ is denoted $X G$ and called the $G$ -invariantsof $X$ .When $X$ is a $G$ -module, $X G$ is the zerothcohomologygroup of $G$ with coefficients in $X$ ,and the higher cohomology groups are thederived functorsof thefunctorof $G$ -invariants.

Fixed points and stabilizer subgroups[edit]

Given $g$ in $G$ and $x$ in $X$ with $g \cdot x = x$ ,it is said that " $x$ is a fixed point of $g$ "or that" $g$ fixes $x$ ".For every $x$ in $X$ ,thestabilizer subgroupof $G$ with respect to $x$ (also called theisotropy grouporlittle group^[10]) is the set of all elements in $G$ that fix $x$ : $G_{x}=\{g\in G:g{\cdot }x=x\}.$ This is asubgroupof $G$ ,though typically not a normal one. The action of $G$ on $X$ isfreeif and only if all stabilizers are trivial. The kernel $N$ of the homomorphism with the symmetric group, $G \to Sym(X)$ ,is given by theintersectionof 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 $g$ be a group element such that $y = g \cdot x$ .Then the two stabilizer groups $G x$ and $G y$ are related by $G y = gG x g -1$ .Proof: by definition, $h \in G y$ if and only if $h \cdot(g \cdot x) = g \cdot x$ .Applying $g -1$ to both sides of this equality yields $(g -1 hg)\cdot x = x$ ;that is, $g -1 hg \in G x$ .An opposite inclusion follows similarly by taking $h \in G x$ and $x = g -1 \cdot y$ .

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

Orbit-stabilizer theoremand Burnside's lemma[edit]

Orbits and stabilizers are closely related. For a fixed $x$ in $X$ ,consider the map $f : G \to X$ given by $g \mapsto g \cdot x$ .By definition the image $f (G)$ of this map is the orbit $G \cdot x$ .The condition for two elements to have the same image is $f(g)=f(h)\iff g{\cdot }x=h{\cdot }x\iff g^{-1}h{\cdot }x=x\iff g^{-1}h\in G_{x}\iff h\in gG_{x}.$ In other words, $f (g) = f (h)$ if and only if $g$ and $h$ lie in the samecosetfor the stabilizer subgroup $G x$ .Thus, thefiber $f -1 ({y})$ of $f$ over any $y$ in $G \cdot x$ is contained in such a coset, and every such coset also occurs as a fiber. Therefore $f$ induces abijectionbetween the set $G / G x$ of cosets for the stabilizer subgroup and the orbit $G \cdot x$ ,which sends $gG x \mapsto g \cdot x$ .^[11]This result is known as theorbit-stabilizer theorem.

If $G$ is finite then the orbit-stabilizer theorem, together withLagrange's theorem,gives $|G\cdot x|=[G\,:\,G_{x}]=|G|/|G_{x}|,$ in other words the length of the orbit of $x$ times the order of its stabilizer is theorder 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 most

k mod 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).

Example:We can use the orbit-stabilizer theorem to count the automorphisms of agraph.Consider thecubical graphas pictured, and let

G

denote itsautomorphismgroup. 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,

| G | = | G \cdot 1 | | G 1 | = 8 | G 1 |

.Applying the theorem now to the stabilizer

G 1

,we can obtain

| 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

2 π /3

,which permutes 2, 4, 5 and 3, 6, 8, and fixes 1 and 7. Thus,

| (G 1) \cdot 2 | = 3

.Applying the theorem a third time gives

| (G 1) 2 | = | ((G 1) 2) \cdot 3 | | ((G 1) 2) 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

| ((G 1) 2) \cdot 3 | = 2

.One also sees that

((G 1) 2) 3

consists only of the identity automorphism, as any element of

G

fi xing 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

| G | = 8 \cdot 3 \cdot 2 \cdot 1 = 48

.

A result closely related to the orbit-stabilizer theorem isBurnside's lemma: $|X/G|={\frac {1}{|G|}}\sum _{g\in G}|X^{g}|,$ 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.

Fi xing a group $G$ ,the set of formal differences of finite $G$ -sets forms a ring called theBurnside ringof $G$ ,where addition corresponds todisjoint union,and multiplication toCartesian product.

Examples[edit]

Thetrivialaction 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 theidentity permutationon $X$ .^[12]
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 ofCayley's theorem– that every group is isomorphic to a subgroup of the symmetric group of permutations of the set $G$ .
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 nontrivialnormal subgroupsof $G$ this induces an isomorphism from $G$ to a subgroup of the permutation group ofdegree $[G : H]$ .
In every group $G$ ,conjugationis 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 ( $x g) h = x gh$ .
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 anautomorphismof $X$ ,given by the action of 1. Similarly, an action of $Z / 2 Z$ on $X$ is equivalent to the data of aninvolutionof $X$ .
The symmetric group $S n$ and its subgroups act on the set ${1,..., n}$ by permuting its elements
Thesymmetry groupof 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.
For acoordinate space $V$ 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 calledscalar multiplication.
The automorphism group of a vector space (orgraph,or group, or ring...) acts on the vector space (or set of vertices of the graph, or group, or ring...).
The general linear group $GL(n, K)$ and its subgroups, particularly itsLie subgroups(including the special linear group $SL(n, K)$ ,orthogonal group $O(n, K)$ ,special orthogonal group $SO(n, K)$ ,andsymplectic group $Sp(n, K)$ ) areLie groupsthat act on the vector space $K n$ .The group operations are given by multiplying the matrices from the groups with the vectors from $K n$ .
The general linear group $GL(n, Z)$ acts on $Z n$ by natural matrix action. The orbits of its action are classified by thegreatest common divisorof coordinates of the vector in $Z n$ .
Theaffine groupactstransitivelyon the points of anaffine 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 anaffine space.
Theprojective linear group $PGL(n + 1, K)$ and its subgroups, particularly its Lie subgroups, which are Lie groups that act on theprojective 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 thecross ratio;theMöbius group $PGL(2, C)$ is of particular interest.
Theisometriesof the plane act on the set of 2D images and patterns, such aswallpaper 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).^{[dubious–discuss]}
The sets acted on by a group $G$ comprise thecategoryof $G$ -sets in which the objects are $G$ -sets and themorphismsare $G$ -set homomorphisms: functions $f : X \to Y$ such that $g \cdot(f (x)) = f (g \cdot x)$ for every $g$ in $G$ .
TheGalois groupof afield extension $L / K$ acts on the field $L$ but has only a trivial action on elements of the subfield $K$ .Subgroups of $Gal(L / K)$ correspond to subfields of $L$ that contain $K$ ,that is, intermediate field extensions between $L$ and $K$ .
The additive group of thereal numbers $(R,+)$ acts on thephase spaceof "well-behaved"systems inclassical mechanics(and in more generaldynamical systems) bytime 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.
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 thepower setof $X$ ,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 largeMathieu groupon a 24-set and in studying symmetry in certain models offinite geometries.
Thequaternionswithnorm1 (theversors), 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; seequaternions and spatial rotation.This is not a faithful action because the quaternion $-1$ leaves all points where they were, as does the quaternion $1$ .
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 anexponential objectin the category of $G$ -sets.

Group actions and groupoids[edit]

The notion of group action can be encoded by theactiongroupoid $G' = G ⋉ X$ associated to the group action. The stabilizers of the action are the vertex groups of the groupoid and the orbits of the action are its components.

Morphisms and isomorphisms betweenG-sets[edit]

If $X$ and $Y$ are two $G$ -sets, amorphismfrom $X$ 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 calledequivariant mapsor $G$ -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 anisomorphism,and the two $G$ -sets $X$ and $Y$ are calledisomorphic;for all practical purposes, isomorphic $G$ -sets are indistinguishable.

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 $G \times S$ ,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.)

With this notion of morphism, the collection of all $G$ -sets forms acategory;this category is aGrothendieck topos(in fact, assuming a classicalmetalogic,thistoposwill even be Boolean).

Variants and generalizations[edit]

We can also consider actions ofmonoidson sets, by using the same two axioms as above. This does not define bijective maps and equivalence relations however. Seesemigroup 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 ofendomorphismsof $X$ .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 obtaingroup representationsin this fashion.

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

In addition tocontinuous actionsof topological groups on topological spaces, one also often considerssmooth actionsof Lie groups onsmooth manifolds,regular actions ofalgebraic groupsonalgebraic varieties,andactionsofgroup schemesonschemes.All of these are examples ofgroup objectsacting on objects of their respective category.

Gallery[edit]

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.

Notes[edit]

Citations[edit]

^Eie & Chang (2010).A Course on Abstract Algebra.p. 144.
^This is done, for example, bySmith (2008).Introduction to abstract algebra.p. 253.
^"Definition:Right Group Action Axioms".Proof Wiki.Retrieved19 December2021.
^Thurston 1997,Definition 3.5.1(iv).
^Kapovich 2009,p. 73.
^Thurston 1980,p. 176.
^Hatcher 2002,p. 72.
^Maskit 1988,II.A.1, II.A.2.
^tom Dieck 1987.
^Procesi, Claudio (2007).Lie Groups: An Approach through Invariants and Representations.Springer Science & Business Media. p. 5.ISBN 9780387289298.Retrieved23 February2017.
^M. Artin,Algebra,Proposition 6.8.4 on p. 179
^Eie & Chang (2010).A Course on Abstract Algebra.p. 145.
^Reid, Miles (2005).Geometry and topology.Cambridge, UK New York: Cambridge University Press. p. 170.ISBN 9780521613255.
^Perrone (2024),pp. 7–9
^Perrone (2024),pp. 36–39
^Perrone (2024),pp. 69–71

References[edit]

Aschbacher, Michael(2000).Finite Group Theory.Cambridge University Press.ISBN 978-0-521-78675-1.MR 1777008.
Dummit, David; Richard Foote (2003).Abstract Algebra(3rd ed.). Wiley.ISBN 0-471-43334-9.
Eie, Minking; Chang, Shou-Te (2010).A Course on Abstract Algebra.World Scientific.ISBN 978-981-4271-88-2.
Hatcher, Allen(2002),Algebraic Topology,Cambridge University Press,ISBN 978-0-521-79540-1,MR 1867354.
Rotman, Joseph (1995).An Introduction to the Theory of Groups.Graduate Texts in Mathematics148(4th ed.). Springer-Verlag.ISBN 0-387-94285-8.
Smith, Jonathan D.H. (2008).Introduction to abstract algebra.Textbooks in mathematics. CRC Press.ISBN 978-1-4200-6371-4.
Kapovich, Michael (2009),Hyperbolic manifolds and discrete groups,Modern Birkhäuser Classics, Birkhäuser, pp. xxvii+467,ISBN 978-0-8176-4912-8,Zbl 1180.57001
Maskit, Bernard (1988),Kleinian groups,Grundlehren der Mathematischen Wissenschaften, vol. 287, Springer-Verlag, pp. XIII+326,Zbl 0627.30039
Perrone, Paolo (2024),Starting Category Theory,World Scientific,doi:10.1142/9789811286018_0005,ISBN 978-981-12-8600-1
Thurston, William (1980),The geometry and topology of three-manifolds,Princeton lecture notes, p. 175, archived fromthe originalon 2020-07-27,retrieved2016-02-08
Thurston, William P. (1997),Three-dimensional geometry and topology. Vol. 1.,Princeton Mathematical Series, vol. 35, Princeton University Press, pp. x+311,Zbl 0873.57001
tom Dieck, Tammo (1987),Transformation groups,de Gruyter Studies in Mathematics, vol. 8, Berlin: Walter de Gruyter & Co., p. 29,doi:10.1515/9783110858372.312,ISBN 978-3-11-009745-0,MR 0889050

External links[edit]

"Action of a group on a manifold",Encyclopedia of Mathematics,EMS Press,2001 [1994]
Weisstein, Eric W."Group Action".MathWorld.

[1] Eie & Chang (2010).A Course on Abstract Algebra.p. 144.

[2] This is done, for example, bySmith (2008).Introduction to abstract algebra.p. 253.

[3] "Definition:Right Group Action Axioms".Proof Wiki.Retrieved19 December2021.

[FOOTNOTEThurston1997Definition_3.5.1(iv)-4] Thurston 1997,Definition 3.5.1(iv).

[FOOTNOTEKapovich2009p._73-5] Kapovich 2009,p. 73.

[FOOTNOTEThurston1980176-6] Thurston 1980,p. 176.

[FOOTNOTEHatcher2002p._72-7] Hatcher 2002,p. 72.

[FOOTNOTEMaskit1988II.A.1,_II.A.2-8] Maskit 1988,II.A.1, II.A.2.

[FOOTNOTEtom_Dieck1987-9] tom Dieck 1987.

[Procesi-10] Procesi, Claudio (2007).Lie Groups: An Approach through Invariants and Representations.Springer Science & Business Media. p. 5.ISBN 9780387289298.Retrieved23 February2017.

[11] M. Artin,Algebra,Proposition 6.8.4 on p. 179

[12] Eie & Chang (2010).A Course on Abstract Algebra.p. 145.

[13] Reid, Miles (2005).Geometry and topology.Cambridge, UK New York: Cambridge University Press. p. 170.ISBN 9780521613255.

[14] Perrone (2024),pp. 7–9

[15] Perrone (2024),pp. 36–39

[16] Perrone (2024),pp. 69–71

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Definition[edit]

Left group action[edit]

Right group action[edit]

Notable properties of actions[edit]

Transitivity properties[edit]

Examples[edit]

Primitive actions[edit]

Topological properties[edit]

Actions of topological groups[edit]

Linear actions[edit]

Orbits and stabilizers[edit]

Invariant subsets[edit]

Fixed points and stabilizer subgroups[edit]

Orbit-stabilizer theoremand Burnside's lemma[edit]

Examples[edit]

Group actions and groupoids[edit]

Morphisms and isomorphisms betweenG-sets[edit]

Variants and generalizations[edit]

Gallery[edit]

See also[edit]

Notes[edit]

Citations[edit]

References[edit]

External links[edit]