Group action

In mathematics, groupsoften useddescribe symmetriesobjects. Thisformalized bynotion ofgroup action: every element ofgroup "acts" likebijective map (or "symmetry") on some set. In this case,groupalso calledtransformation group ofset.

Tablecontents

1 Definition 2 Examples 3 Typesactions 4 Orbitsstabilizers 5 Morphismsisomorphisms between G-sets 6 Generalizations

Definition

If G isgroupX isset, then(left) group actionG on X isbinary function G × X -> X (whereimagegGxXwritten as g.x) which satisfiesfollowing two axioms:

1. g.(h.x) = (gh).xall g, hGxX.
2. e.x = xevery xX; here e denotesidentity elementG.

From these two axioms,follows thatevery gG,function which maps xXg.x isbijective map from XX. Therefore, one may alternativelyequivalently definegroup actionG on X asgroup homomorphism G -> Sym(X), where Sym(X) denotesgroupall bijective maps from XX.

Ifgroup action G × X -> Xgiven, we also say that G acts onset X or X isG-set.

In complete analogy, one can defineright group actionG on X asfunction X × G -> X bytwo axioms (x.g).h = x.(gh)x.e = x. Insequel, we consider only left group actions.

Examples

• Every group G acts on Gtwo natural ways: g.x = (gx)all xG, or g.x = (gxg ^-1)all xG.
• The symmetric group S_nits subgroups act onset { 1, ... , n } by permutating its elements.
• The symmetry group ofpolyhedron acts onsetverticesthat polyhedron.
• The symmetry groupany geometrical object acts onsetpointsthat object.
• The automorphism group ofvector space (or graph, or group, or ring...) acts onvector space (or setvertices ofgraph, or group, or ring...).
• The Lie groups Gl(n,R), SL(n,R)O(n,R) act on Rⁿ.
• The Galois group offield extension E/F acts onbigger field E. So does every subgroup ofGalois group.
• The additive group ofreal numbers (R, +) acts onphase space"well-behaved" systemsclassical mechanics (andmore general dynamical systems): if tin Rxinphase space, then x describesstate ofsystem,t.xdefinedbestate ofsystem t seconds later if tpositive or -t seconds ago if tnegative.

Typesactions

The actionG on Xcalled

• transitive ifany two x, yX there exists an gG such that g.x = y;
• simply transitive ifany two x, yX there exists precisely one gG such that g.x = y.
• faithful (or effective) ifany two different g, hG there exists an xX such that g.x ≠ h.x;
• free ifany two different g, hGall xX we have g.x ≠ h.x;

Every free action onnon-empty setfaithful. A group G that acts faithfully onset Xisomorphic topermutation group on X. An actionregular ifonly if ittransitivefree.

Orbitsstabilizers

If we define N = {gG : g.x = xall xX}, then N isnormal subgroupG andfactor group G/N acts faithfully on X by setting (gN).x = g.x. The actionG on Xfaithful ifonly if N = {e}.

If Y issubsetX, we write GY forset { g.y : yYgG}. We callsubset Y invariant under G if GY = Y (whichequivalentGY ⊆ Y). In that case, G also operates on Y. The subset Ycalled fixed under G if g.y = yall gGall yY. Every subset that's fixed under Galso invariant under G, but not vice versa.

Any operationG on X defines an equivalence relation on X: two elements xycalled equivalent if there existsgGg.x = y. The equivalence classx under this equivalence relationgiven byset Gx = { g.x : gG } whichalso calledorbitx. The elements xyequivalent ifonly if their orbits aresame: Gx = Gy. Every orbitan invariant subsetX on which G acts transitively. The actionG on Xtransitive ifonly if all elementsequivalent, meaning that thereonly one orbit. The setall orbitswritten as X/G.

For every xX, we define G_x = { gG : g.x = x }. This issubgroupG,itcalledstabilizerx or isotropy subgroup at x. The actionG on Xfree ifonly if all stabilizers consist only ofidentity element.

There isnatural bijection betweensetall left cosets ofsubgroup G_x andorbitx, given by hG_x |-> h.x. Therefore, |Gx| = [G : G_x], and so

This result, known asorbit-stabilizer theorem,especially useful if GXfinite, because thencan be employedcounting arguments. A related resultBurnside's lemma:

where r isnumberorbits,X^g issetpoints fixed by g. This result toomainlyuse when GXfinite, whencan be interpreted as follows:numberorbitsequal toaverage numberpoints fixed per group element.

Morphismsisomorphisms between G-sets

If XYtwo G-sets, we definemorphism from XY to befunction f : X -> Y such that f(g.x) = g.f(x)all gGall xX. If suchfunction fbijective, then its inversealsomorphism,we call f an isomorphism andtwo G-sets XYcalled isomorphic;all practical purposes, theyindistinguishablethis case.

Some example isomorphisms:

• Every regular G actionisomorphic toactionG on G given by left multiplication.
• Every free G actionisomorphicG×S, where Ssome setG acts by left multiplication onfirst coordinate.
• Every transitive G actionisomorphicleft multiplication by G onsetleft cosetssome subgroup HG.

With this notionmorphism,collectionall G-sets formscategory; this category istopos.

Generalizations

One often considers continuous group actions:group G istopological group, X istopological space, andmap G × X → Xcontinuousrespect toproduct topologyG × X. The space Xalso calledG-spacethis case. Thisindeedgeneralization, since every group can be consideredtopological group by usingdiscrete topology. Allconcepts introduced above still workthis context, however we define morphisms between G-spacesbe continuous maps compatible withactionG. The above statements about isomorphismsregular, freetransitive actionsno longer validcontinuous group actions.

One can also consider actionsmonoids on sets, by usingsame two axioms as above. This does not define bijective mapsequivalence relations however.

Insteadactions on sets, one can define actionsgroupsmonoids on objectsan arbitrary category: startan object Xsome category,then define an action on X asmonoid homomorphism intomonoidendomorphismsX. If X has an underlying set, then all definitionsfacts stated above can be carried over. For example, if we takecategoryvector spaces, we obtain group representationsthis fashion.

One can viewgroup G ascategory withsingle objectwhich every morphisminvertible. A group actionthen nothing butfunctor from G tocategorysets, andgroup representation isfunctor from G tocategoryvector spaces. In analogy, an action ofgroupoid isfunctor fromgroupoid tocategorysets orsome other category.