8097:
8085:
820:
3772:
59:
5710:
5361:
1559:
869:
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
6884:. 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).
2436:
in 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
6076:
8096:
2256:
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
5196:
8084:
2007:
1276:
2160:
1746:
3947:
4767:
1873:
1138:
5623:
2480:
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
1435:
1927:
1196:
2089:
1680:
504:
479:
442:
5960:
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
1446:
806:
1038:
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
7979:
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
7603:
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.
5917:
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
5837:
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
8765:
5979:
364:
8685:
8620:
8570:
8551:
8501:
8477:
8431:
2232:, 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
8760:
5749:, and this action is transitive as can be seen by composing rotations about the center of the cube. Thus, by the orbit-stabilizer theorem,
6102:
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.
5515:
314:
6569:
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.
3889:
799:
309:
8532:
8458:
8363:
8301:
4704:
7951:
Instead of actions on sets, we can define actions of groups and monoids on objects of an arbitrary category: start with an object
7430:
5521:
725:
8291:
8133:
7944:
on sets, by using the same two axioms as above. This does not define bijective maps and equivalence relations however. See
1938:
1207:
8719:
8338:
6361:
792:
2100:
1686:
923:
If a group acts on a structure, it will usually also act on objects built from that structure. For example, the group of
8177:
8159:
5722:
8714:
3776:
3065:
2974:
2634:
1572:. The second axiom then states that the function composition is compatible with the group multiplication; they form a
409:
223:
8709:
2512:. The smallest sets on which faithful actions can be defined for these groups are of size 5, 7, and 16 respectively.
2260:
both a left action and a right action on the group itself—multiplication on the left and on the right, respectively.
4792:
4424:
4060:
3744:
1837:
1102:
8630:
8017:
4813:
4334:
3390:
2839:
1350:
846:
6675:, or group, or ring ...) acts on the vector space (or set of vertices of the graph, or group, or ring ...).
5700:
This result is especially useful since it can be employed for counting arguments (typically in situations where
5356:{\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}.}
6533:
607:
341:
218:
106:
28:
8123:
8038:
6850:. This is a quotient of the action of the general linear group on projective space. Particularly notable is
6822:
6786:
3676:
3411:
3016:
966:
of the group. In the case of a finite-dimensional vector space, it allows one to identify many groups with
8027:
6894:
3361:
2257:
1094:
871:
858:
834:
819:
757:
547:
8755:
6665:
3351:
2428:
For example, the action of any group on itself by left multiplication is free. This observation implies
631:
1398:
8118:
8102:
Orbit of a fundamental spherical triangle (marked in red) under action of the full icosahedral group.
8066:
7980:
7359:
6810:, and the subgroup V of the affine group (that is, a vector space) has transitive and free (that is,
5632:
5116:
3980:
3736:
3724:
2948:
2931:
1569:
971:
963:
911:
854:
571:
559:
177:
111:
8591:, Grundlehren der Mathematischen Wissenschaften, vol. 287, Springer-Verlag, pp. XIII+326,
8090:
Orbit of a fundamental spherical triangle (marked in red) under action of the full octahedral group.
1891:
1160:
8672:, de Gruyter Studies in Mathematics, vol. 8, Berlin: Walter de Gruyter & Co., p. 29,
8058:
7063:
6758:. The group operations are given by multiplying the matrices from the groups with the vectors from
6266:
5973:
5412:
3271:
2429:
1573:
1060:
999:
986:
850:
146:
41:
2060:
1651:
487:
462:
425:
8419:
7921:
7559:
6374:
6269:– that every group is isomorphic to a subgroup of the symmetric group of permutations of the set
5736:
3970:
2996:
2838:(that is, subgroups of a finite symmetric group whose action is 2-transitive) and more generally
2835:
924:
901:
131:
103:
3656:. Contrary to what the name suggests, this is a weaker property than continuity of the action.
8770:
8728:
8681:
8616:
8566:
8547:
8528:
8497:
8473:
8454:
8427:
8359:
8297:
8054:
7067:
6203:
6132:
4023:
3786:
3771:
3423:
3275:
3034:
2908:
1025:
944:
892:
702:
536:
379:
273:
8193:
1554:{\displaystyle \alpha _{g}(\alpha _{h}(x))=(\alpha _{g}\circ \alpha _{h})(x)=\alpha _{gh}(x)}
8673:
8655:
8608:
8592:
8576:
8042:
8007:
7945:
7071:
6837:
6734:
6707:
6576:
6345:
4550:
4209:
3740:
3712:
3451:
2958:
1064:
687:
679:
671:
663:
655:
643:
583:
523:
513:
355:
297:
172:
141:
8695:
8654:, Princeton Mathematical Series, vol. 35, Princeton University Press, pp. x+311,
8511:
8441:
8691:
8659:
8596:
8580:
8507:
8437:
8050:
7993:
7917:
7722:
7348:
6980:
6881:
5047:
5043:
4566:
3755:
3282:
is wandering and free. Such actions can be characterized by the following property: every
1011:
952:
948:
928:
771:
764:
750:
707:
595:
518:
348:
262:
202:
82:
17:
6863:
3804:(of order 60/12 = 5) is naturally identified with the 5 tetrahedra – the coset
3389:. For a properly discontinuous action, cocompactness is equivalent to compactness of the
2947:
of non-zero vectors is transitive, but not 2-transitive (similarly for the action of the
6836:
and its subgroups, particularly its Lie subgroups, which are Lie groups that act on the
6265:. This action is free and transitive (regular), and forms the basis of a rapid proof of
3337:, and the largest subset on which the action is freely discontinuous is then called the
3156:
of the action is the set of all points of discontinuity. Equivalently it is the largest
6858:, the symmetries of the projective line, which is sharply 3-transitive, preserving the
6566:
6128:
4213:
3993:
3554:
3279:
2241:
862:
778:
714:
404:
384:
321:
286:
207:
197:
182:
167:
121:
98:
6572:
The symmetry group of any geometrical object acts on the set of points of that object.
8749:
8491:
8487:
8128:
7964:
7344:
6118:
5726:
5447:
is contained in such a coset, and every such coset also occurs as a fiber. Therefore
3785:
of order 60, while the stabilizer of a single chosen tetrahedron is the (rotational)
3193:
3181:
3044:
697:
619:
453:
326:
192:
5635:. In particular that implies that the orbit length is a divisor of the group order.
8070:
8062:
7059:
6976:
6815:
6807:
6799:
6691:
6672:
6507:
2461:
2336:
959:
824:
552:
251:
240:
187:
162:
157:
116:
50:
8034:
is a functor from the groupoid to the category of sets or to some other category.
5709:
4449:
if and only if all elements are equivalent, meaning that there is only one orbit.
8731:
8667:
3432:
a topological space on which it acts by homeomorphisms. The action is said to be
2192:
The difference between left and right actions is in the order in which a product
7753:
7055:
7044:
6859:
2962:
1039:
1035:
842:
8612:
3951:
The defining properties of a group guarantee that the set of orbits of (points
874:
acts also on various related structures; for example, the above rotation group
8677:
8113:
7355:
3760:
3708:
3508:
3012:
940:
936:
931:
and also on the figures drawn in it; in particular, it acts on the set of all
719:
447:
6814:) action on these points; indeed this can be used to give a definition of an
2834:
this is often called double, respectively triple, transitivity. The class of
1614:
entirely, and to replace it with either a dot, or with nothing at all. Thus,
8736:
7925:
7283:
6749:
3666:
2433:
2352:
1778:
990:
967:
861:
around a point in the plane. It is often useful to consider the group as an
540:
27:
This article is about the mathematical concept. For the sociology term, see
7548:-set has the property that its fixed points correspond to equivariant maps
7144:
acts on the set of real functions of a real variable in various ways, with
3011:
apart from the trivial partitions (the partition in a single piece and its
2961:
of a
Euclidean space is not transitive on nonzero vectors but it is on the
58:
8046:
8031:
7586:
7578:
6910:
6877:
6785:
by natural matrix action. The orbits of its action are classified by the
6403:. An exponential notation is commonly used for the right-action variant:
4770:
4535:
1340:
1034:
elements by permuting the elements of the set. Although the group of all
932:
77:
5721:
We can use the orbit-stabilizer theorem to count the automorphisms of a
1645:, especially when the action is clear from context. The axioms are then
7997:
7900:
can be taken to be the stabilizer group of any element of the original
4570:
3231:. This is strictly stronger than wandering; for instance the action of
419:
333:
8636:
8041:
of topological groups on topological spaces, one also often considers
5042:
The above says that the stabilizers of elements in the same orbit are
3547:
is compact. In particular, this is equivalent to proper discontinuity
7941:
7363:
4231:
1144:
7739:
The composition of two morphisms is again a morphism. If a morphism
7343:. This is useful, for instance, in studying the action of the large
4208:. The coinvariant terminology and notation are used particularly in
8358:. Cambridge, UK New York: Cambridge University Press. p. 170.
7929:
5708:
5397:
4063:
if and only if it has exactly one orbit, that is, if there exists
4038:
are equivalent if and only if their orbits are the same, that is,
3770:
818:
1608:
With the above understanding, it is very common to avoid writing
7745:
is bijective, then its inverse is also a morphism. In this case
8030:
between the group action functors. In analogy, an action of a
6880:
of the plane act on the set of 2D images and patterns, such as
5845:, which permutes 2, 4, 5 and 3, 6, 8, and fixes 1 and 7. Thus,
8565:, Modern Birkhäuser Classics, Birkhäuser, pp. xxvii+467,
8293:
Lie Groups: An
Approach through Invariants and Representations
7992:
as a category with a single object in which every morphism is
5954:
consists only of the identity automorphism, as any element of
4845:
is trivial, the action is said to be faithful (or effective).
4175:, while in algebraic situations it may be called the space of
878:
also on triangles by transforming triangles into triangles.
6071:{\displaystyle |X/G|={\frac {1}{|G|}}\sum _{g\in G}|X^{g}|,}
5972:
A result closely related to the orbit-stabilizer theorem is
4167:
of the action. In geometric situations it may be called the
2601:) if it is both transitive and free. This means that given
5514:
is finite then the orbit-stabilizer theorem, together with
5193:. The condition for two elements to have the same image is
3779:, the symmetry group is the (rotational) icosahedral group
3179:
is wandering. In a dynamical context this is also called a
1787:. Therefore, one may equivalently define a group action of
1576:. This axiom can be shortened even further, and written as
837:
by 0°, 120° and 240° acts on the set of the three vertices.
7996:. A (left) group action is then nothing but a (covariant)
7437:
leaves all points where they were, as does the quaternion
7347:
on a 24-set and in studying symmetry in certain models of
6998:
but has only a trivial action on elements of the subfield
5056:(that is, the set of all conjugates of the subgroup). Let
4194:, by contrast with the invariants (fixed points), denoted
5132:
Orbits and stabilizers are closely related. For a fixed
2047:
when the action being considered is clear from context)
7433:. This is not a faithful action because the quaternion
5046:
to each other. Thus, to each orbit, we can associate a
4795:
if and only if all stabilizers are trivial. The kernel
4216:, which use the same superscript/subscript convention.
8635:, Princeton lecture notes, p. 175, archived from
3267:
is wandering and free but not properly discontinuous.
2857:
when the action on tuples without repeated entries in
2827:
of tuples without repeated entries is transitive. For
2425:. This is a much stronger property than faithfulness.
6803:
5982:
5524:
5199:
4707:
3892:
3111:
is called a point of discontinuity for the action of
2103:
2063:
2002:{\displaystyle \alpha (\alpha (x,g),h)=\alpha (x,gh)}
1941:
1894:
1840:
1751:
From these two axioms, it follows that for any fixed
1689:
1654:
1449:
1401:
1392:. The identity and compatibility relations then read
1271:{\displaystyle \alpha (g,\alpha (h,x))=\alpha (gh,x)}
1210:
1163:
1105:
490:
465:
428:
8296:. Springer Science & Business Media. p. 5.
7910:
With this notion of morphism, the collection of all
6516:, given by the action of 1. Similarly, an action of
5472:
of cosets for the stabilizer subgroup and the orbit
5010:. An opposite inclusion follows similarly by taking
4779:, though typically not a normal one. The action of
3715:, i.e. action which are smooth on the whole space.
2155:{\displaystyle (x{\cdot }g){\cdot }h=x{\cdot }(gh)}
1781:, with inverse bijection the corresponding map for
1741:{\displaystyle g{\cdot }(h{\cdot }x)=(gh){\cdot }x}
6070:
5617:
5355:
4761:
3941:
3333:. Actions with this property are sometimes called
2154:
2083:
2001:
1921:
1867:
1740:
1674:
1553:
1429:
1270:
1190:
1132:
498:
473:
436:
7851:by left multiplication on the first coordinate. (
7583:The notion of group action can be encoded by the
7028:, that is, intermediate field extensions between
3942:{\displaystyle G{\cdot }x=\{g{\cdot }x:g\in G\}.}
2473:cannot act faithfully on a set of size less than
8652:Three-dimensional geometry and topology. Vol. 1.
8073:acting on objects of their respective category.
7411:is a counterclockwise rotation through an angle
2621:in the definition of transitivity is unique. If
8010:, and a group representation is a functor from
7957:of some category, and then define an action on
7876:action is isomorphic to left multiplication by
2460:. This is not always the case, for example the
34:Transformations induced by a mathematical group
4801:of the homomorphism with the symmetric group,
4762:{\displaystyle G_{x}=\{g\in G:g{\cdot }x=x\}.}
6671:The automorphism group of a vector space (or
5966:| = 8 ⋅ 3 ⋅ 2 ⋅ 1 = 48
5784:. Applying the theorem now to the stabilizer
2821:. In other words the action on the subset of
2627:is acted upon simply transitively by a group
1868:{\displaystyle \alpha \colon X\times G\to X,}
1133:{\displaystyle \alpha \colon G\times X\to X,}
800:
8:
8632:The geometry and topology of three-manifolds
7963:as a monoid homomorphism into the monoid of
6506:uniquely determines and is determined by an
4753:
4721:
3933:
3907:
2413:. In other words, no non-trivial element of
1349:, so that, instead, one has a collection of
7882:on the set of left cosets of some subgroup
6202:; that is, every group element induces the
4565:, and the higher cohomology groups are the
6360:to a subgroup of the permutation group of
6111:, the set of formal differences of finite
5860:. Applying the theorem a third time gives
5625:in other words the length of the orbit of
5330:
5326:
5296:
5292:
5261:
5257:
5231:
5227:
3278:of a locally simply connected space on an
2240:can be considered as a left action of its
807:
793:
245:
71:
36:
8277:
7779:; for all practical purposes, isomorphic
7108:is defined to be the state of the system
6060:
6054:
6045:
6033:
6021:
6013:
6007:
5999:
5991:
5983:
5981:
5631:times the order of its stabilizer is the
5607:
5601:
5592:
5587:
5582:
5574:
5562:
5557:
5553:
5539:
5525:
5523:
5344:
5320:
5301:
5278:
5266:
5249:
5235:
5198:
5115:. A maximal orbit type is often called a
4739:
4712:
4706:
3913:
3896:
3891:
2135:
2121:
2110:
2102:
2067:
2062:
1940:
1893:
1839:
1730:
1704:
1693:
1688:
1658:
1653:
1533:
1508:
1495:
1467:
1454:
1448:
1406:
1400:
1209:
1162:
1104:
492:
491:
489:
467:
466:
464:
430:
429:
427:
8563:Hyperbolic manifolds and discrete groups
8241:
8229:
8217:
5618:{\displaystyle |G\cdot x|==|G|/|G_{x}|,}
3810:corresponds to the tetrahedron to which
3753:-invariant submodules. It is said to be
2842:is well-studied in finite group theory.
2727:with pairwise distinct entries (that is
8546:. Textbooks in mathematics. CRC Press.
8521:An Introduction to the Theory of Groups
8402:
8390:
8378:
8253:
8150:
8080:
7136:The additive group of the real numbers
3127:such that there are only finitely many
3073:such that there are only finitely many
363:
129:
39:
8265:
8194:"Definition:Right Group Action Axioms"
7798:action is isomorphic to the action of
6290:, left multiplication is an action of
6221:, left multiplication is an action of
5666:elements. Since each orbit has either
4974:to both sides of this equality yields
4427:. Conversely, any invariant subset of
4411:Every orbit is an invariant subset of
3711:. There is a well-developed theory of
365:Classification of finite simple groups
8468:Eie, Minking; Chang, Shou-Te (2010).
8449:Dummit, David; Richard Foote (2003).
7857:can be taken to be the set of orbits
7417:about an axis given by a unit vector
7366:), as a multiplicative group, act on
7276:, we can define an induced action of
7098:describes a state of the system, and
4583:Fixed points and stabilizer subgroups
3511:. This means that given compact sets
2871:The action of the symmetric group of
1878:that satisfies the analogous axioms:
1339:It can be notationally convenient to
7:
7536:" indicates right multiplication by
6694:(including the special linear group
6690:and its subgroups, particularly its
6090:. This result is mainly of use when
4433:is a union of orbits. The action of
3735:acts by linear transformations on a
3683:for the action is the set of points
7607:Morphisms and isomorphisms between
5713:Cubical graph with vertices labeled
4396:. Every subset that is fixed under
2453:) acts faithfully on a set of size
4089:. This is the case if and only if
4026:under this relation; two elements
2957:is at least 2). The action of the
25:
8710:"Action of a group on a manifold"
8527:(4th ed.). Springer-Verlag.
6554:and its subgroups act on the set
6354:this induces an isomorphism from
4886:. Then the two stabilizer groups
3794:of order 12, and the orbit space
3365:if there exists a compact subset
1143:that satisfies the following two
8544:Introduction to abstract algebra
8523:. Graduate Texts in Mathematics
8179:Introduction to abstract algebra
8095:
8083:
7940:We can also consider actions of
7431:quaternions and spatial rotation
6789:of coordinates of the vector in
6532:is equivalent to the data of an
6127:, where addition corresponds to
4689:) is the set of all elements in
3864:can be moved by the elements of
3564:if there exists a neighbourhood
1430:{\displaystyle \alpha _{e}(x)=x}
57:
8766:Representation theory of groups
8069:. All of these are examples of
7924:(in fact, assuming a classical
6903:-sets in which the objects are
3747:if there are no proper nonzero
2351:corresponding to the action is
2218:second. Because of the formula
8496:, Cambridge University Press,
8426:. Cambridge University Press.
8174:This is done, for example, by
6919:-set homomorphisms: functions
6441:, conjugation is an action of
6117:-sets forms a ring called the
6084:is the set of points fixed by
6061:
6046:
6022:
6014:
6000:
5984:
5608:
5593:
5583:
5575:
5568:
5547:
5540:
5526:
5501:. This result is known as the
5327:
5293:
5258:
5228:
5224:
5218:
5209:
5203:
5064:denote the conjugacy class of
2682:elements, and for any pair of
2345:to the group of bijections of
2149:
2140:
2118:
2104:
1996:
1981:
1972:
1963:
1951:
1945:
1922:{\displaystyle \alpha (x,e)=x}
1910:
1898:
1856:
1727:
1718:
1712:
1698:
1548:
1542:
1523:
1517:
1514:
1488:
1482:
1479:
1473:
1460:
1418:
1412:
1265:
1250:
1241:
1238:
1226:
1214:
1191:{\displaystyle \alpha (e,x)=x}
1179:
1167:
1121:
726:Infinite dimensional Lie group
1:
8650:Thurston, William P. (1997),
8542:Smith, Jonathan D.H. (2008).
8327:, Proposition 6.8.4 on p. 179
7810:given by left multiplication.
7785:-sets are indistinguishable.
6887:The sets acted on by a group
6752:that act on the vector space
4872:be a group element such that
3816:sends the chosen tetrahedron.
3406:Actions of topological groups
3206:there are only finitely many
3005:preserved by all elements of
2264:Notable properties of actions
2210:second. For a right action,
1795:as a group homomorphism from
8470:A Course on Abstract Algebra
8340:A Course on Abstract Algebra
8161:A Course on Abstract Algebra
7936:Variants and generalizations
7579:Groupoid § Group action
7092:is in the phase space, then
5745:acts on the set of vertices
5676:elements, there are at most
5400:for the stabilizer subgroup
2084:{\displaystyle x{\cdot }e=x}
1675:{\displaystyle e{\cdot }x=x}
865:, and to say that one has a
499:{\displaystyle \mathbb {Z} }
474:{\displaystyle \mathbb {Z} }
437:{\displaystyle \mathbb {Z} }
8761:Group actions (mathematics)
8715:Encyclopedia of Mathematics
7788:Some example isomorphisms:
7573:Group actions and groupoids
7558:; more generally, it is an
7016:correspond to subfields of
6721:, special orthogonal group
4204:while the invariants are a
3777:compound of five tetrahedra
3743:, the action is said to be
3117:if there is an open subset
2975:primitive permutation group
2635:principal homogeneous space
2274:be a group acting on a set
1320:together with an action of
224:List of group theory topics
8787:
8629:Thurston, William (1980),
8613:10.1142/9789811286018_0005
8561:Kapovich, Michael (2009),
7576:
7429:is the same rotation; see
7372:: for any such quaternion
7043:The additive group of the
6873:is of particular interest.
5648:be a group of prime order
5172:. By definition the image
4022:. The orbits are then the
3852:is the set of elements in
3722:
3409:
2972:
2840:multiply transitive groups
1371:, with one transformation
935:. Similarly, the group of
26:
18:Orbit equivalence relation
8678:10.1515/9783110858372.312
8666:tom Dieck, Tammo (1987),
8290:Procesi, Claudio (2007).
8134:Young–Deruyts development
8018:category of vector spaces
6767:The general linear group
6678:The general linear group
6664:is a group action called
6562:by permuting its elements
5183:of this map is the orbit
4200:: the coinvariants are a
4149:(or, less frequently, as
4127:The set of all orbits of
3628:The action is said to be
3457:The action is said to be
2889:up to the cardinality of
1799:into the symmetric group
8605:Starting Category Theory
8587:Maskit, Bernard (1988),
8337:Eie & Chang (2010).
8158:Eie & Chang (2010).
7819:action is isomorphic to
7264:Given a group action of
6131:, and multiplication to
5503:orbit-stabilizer theorem
5125:Orbit-stabilizer theorem
4931:. Proof: by definition,
4402:is also invariant under
3763:of irreducible actions.
3646:is continuous for every
3169:such that the action of
3101:More generally, a point
2863:is sharply transitive.
2487:, the icosahedral group
2374:) if the statement that
2214:acts first, followed by
2206:acts first, followed by
342:Elementary abelian group
219:Glossary of group theory
29:group action (sociology)
8603:Perrone, Paolo (2024),
8519:Rotman, Joseph (1995).
8453:(3rd ed.). Wiley.
8220:, Definition 3.5.1(iv).
8124:Measurable group action
7932:will even be Boolean).
7160:equal to, for example,
6823:projective linear group
6787:greatest common divisor
6344:contains no nontrivial
3679:, then the subspace of
3677:differentiable manifold
3412:Continuous group action
3154:domain of discontinuity
2877:is transitive, in fact
2516:Transitivity properties
2280:. The action is called
1807:of all bijections from
1382:for each group element
1316:(from the left). A set
1312:is then said to act on
8028:natural transformation
7720:-sets are also called
6072:
5714:
5619:
5357:
4763:
4501:. The set of all such
4408:, but not conversely.
3943:
3817:
3767:Orbits and stabilizers
3759:if it decomposes as a
3450:is continuous for the
3190:properly discontinuous
3023:Topological properties
2540:if for any two points
2432:that any group can be
2156:
2085:
2003:
1923:
1869:
1742:
1676:
1555:
1431:
1272:
1192:
1134:
838:
758:Linear algebraic group
500:
475:
438:
8669:Transformation groups
8356:Geometry and topology
8049:, regular actions of
8020:. A morphism between
7981:group representations
7920:; this category is a
7066:(and in more general
6666:scalar multiplication
6296:on the set of cosets
6073:
5936:. One also sees that
5729:as pictured, and let
5712:
5620:
5358:
5122:
4764:
4559:with coefficients in
4307:(which is equivalent
4159:), and is called the
3983:is defined by saying
3944:
3774:
3352:locally compact space
3344:An action of a group
3015:, the partition into
2502:and the cyclic group
2398:already implies that
2358:The action is called
2202:. For a left action,
2157:
2086:
2004:
1924:
1870:
1763:to itself which maps
1743:
1677:
1556:
1432:
1273:
1193:
1135:
910:to some group (under
822:
501:
476:
439:
8607:, World Scientific,
8472:. World Scientific.
8354:Reid, Miles (2005).
8119:Group with operators
7986:We can view a group
6806:on the points of an
6590:with group of units
6545:The symmetric group
6204:identity permutation
6152:action of any group
5980:
5706:is finite as well).
5696:-invariant elements.
5522:
5197:
5128:and Burnside's lemma
5117:principal orbit type
4705:
4631:is a fixed point of
4133:under the action of
4059:The group action is
3981:equivalence relation
3963:under the action of
3890:
3725:Group representation
3335:freely discontinuous
3292:has a neighbourhood
3272:deck transformations
3162:-stable open subset
2951:if the dimension of
2949:special linear group
2932:general linear group
2919:-transitive but not
2907:, the action of the
2883:-transitive for any
2633:then it is called a
2335:. Equivalently, the
2101:
2061:
1939:
1892:
1838:
1759:, the function from
1687:
1652:
1629:can be shortened to
1570:function composition
1447:
1399:
1208:
1161:
1103:
972:general linear group
958:A group action on a
925:Euclidean isometries
914:) of functions from
912:function composition
855:function composition
488:
463:
426:
8424:Finite Group Theory
8420:Aschbacher, Michael
8055:algebraic varieties
7562:in the category of
7474:whose elements are
7064:classical mechanics
6338:. In particular if
5144:, consider the map
4860:be two elements in
4816:of the stabilizers
4665:stabilizer subgroup
4625:, it is said that "
4024:equivalence classes
3632:if the orbital map
3630:strongly continuous
2836:2-transitive groups
2031:often shortened to
1574:commutative diagram
987:invertible matrices
985:, the group of the
955:of the polyhedron.
857:; for example, the
132:Group homomorphisms
42:Algebraic structure
8729:Weisstein, Eric W.
8493:Algebraic Topology
8039:continuous actions
7922:Grothendieck topos
7560:exponential object
7480:-equivariant maps
7462:, there is a left
6992:acts on the field
6882:wallpaper patterns
6068:
6044:
5715:
5633:order of the group
5615:
5516:Lagrange's theorem
5353:
5084:if the stabilizer
4812:, is given by the
4759:
3939:
3818:
3693:such that the map
3037:and the action of
2934:of a vector space
2930:The action of the
2591:sharply transitive
2152:
2081:
1999:
1919:
1865:
1821:right group action
1815:Right group action
1738:
1672:
1551:
1427:
1268:
1188:
1130:
1074:is a set, then a (
902:group homomorphism
839:
833:consisting of the
608:Special orthogonal
496:
471:
434:
315:Lagrange's theorem
8687:978-3-11-009745-0
8622:978-981-12-8600-1
8572:978-0-8176-4912-8
8553:978-1-4200-6371-4
8503:978-0-521-79540-1
8479:978-981-4271-88-2
8433:978-0-521-78675-1
8268:, II.A.1, II.A.2.
8045:of Lie groups on
7983:in this fashion.
7870:Every transitive
7504:-action given by
7349:finite geometries
7319:for every subset
7114:seconds later if
7068:dynamical systems
6447:on conjugates of
6133:Cartesian product
6029:
6027:
5911:. Any element of
5831:. Any element of
5773:| = 8 |
5686:orbits of length
5070:. Then the orbit
5050:of a subgroup of
4681:(also called the
4327:also operates on
4321:). In that case,
4220:Invariant subsets
3979:. The associated
3820:Consider a group
3787:tetrahedral group
3713:Lie group actions
3560:It is said to be
3424:topological group
3276:fundamental group
3035:topological space
2969:Primitive actions
2909:alternating group
2777:) there exists a
2585:simply transitive
2419:fixes a point of
2165:
2164:
2012:
2011:
1281:
1280:
1051:Left group action
817:
816:
392:
391:
274:Alternating group
231:
230:
16:(Redirected from
8778:
8742:
8741:
8723:
8698:
8662:
8646:
8645:
8644:
8625:
8599:
8583:
8557:
8538:
8514:
8483:
8464:
8451:Abstract Algebra
8445:
8406:
8405:, pp. 69–71
8400:
8394:
8393:, pp. 36–39
8388:
8382:
8376:
8370:
8369:
8351:
8345:
8344:
8334:
8328:
8321:
8315:
8314:
8312:
8310:
8287:
8281:
8275:
8269:
8263:
8257:
8251:
8245:
8239:
8233:
8227:
8221:
8215:
8209:
8208:
8206:
8204:
8190:
8184:
8183:
8172:
8166:
8165:
8155:
8099:
8087:
8051:algebraic groups
8047:smooth manifolds
8026:-sets is then a
8025:
8015:
8008:category of sets
8005:
7991:
7978:
7972:
7962:
7956:
7946:semigroup action
7915:
7905:
7899:
7893:
7887:
7881:
7875:
7866:
7856:
7850:
7840:
7835:is some set and
7834:
7828:
7818:
7809:
7803:
7797:
7784:
7774:
7768:
7762:
7750:
7744:
7731:
7723:equivariant maps
7719:
7713:
7707:
7701:
7695:
7689:
7662:
7648:
7642:
7632:
7626:
7620:
7602:
7567:
7557:
7547:
7541:
7535:
7528:
7503:
7498:, and with left
7497:
7479:
7473:
7467:
7461:
7455:
7449:
7440:
7436:
7428:
7422:
7416:
7410:
7390:
7371:
7342:
7336:
7330:
7324:
7318:
7291:
7281:
7275:
7269:
7260:
7245:
7231:
7213:
7199:
7188:
7174:
7159:
7143:
7132:
7126:
7119:
7113:
7107:
7097:
7091:
7085:
7079:
7072:time translation
7053:
7039:
7033:
7027:
7021:
7015:
7003:
6997:
6991:
6971:
6965:
6959:
6932:
6918:
6908:
6902:
6892:
6872:
6857:
6849:
6838:projective space
6835:
6794:
6784:
6778:
6763:
6757:
6747:
6735:symplectic group
6732:
6720:
6708:orthogonal group
6705:
6689:
6663:
6610:
6596:
6589:
6583:
6577:coordinate space
6561:
6553:
6541:
6531:
6525:
6515:
6505:
6499:
6490:
6484:
6478:
6472:
6466:
6452:
6446:
6440:
6434:
6425:
6416:; it satisfies (
6415:
6402:
6388:
6382:
6377:is an action of
6372:
6359:
6353:
6346:normal subgroups
6343:
6337:
6331:
6325:
6319:
6305:
6295:
6289:
6283:
6274:
6267:Cayley's theorem
6264:
6258:
6252:
6246:
6232:
6226:
6220:
6211:
6201:
6195:
6189:
6183:
6177:
6163:
6157:
6150:
6149:
6126:
6116:
6110:
6101:
6095:
6089:
6083:
6077:
6075:
6074:
6069:
6064:
6059:
6058:
6049:
6043:
6028:
6026:
6025:
6017:
6008:
6003:
5995:
5987:
5974:Burnside's lemma
5967:
5965:
5959:
5953:
5935:
5933:
5916:
5910:
5908:
5890:
5875:
5859:
5857:
5844:
5836:
5830:
5828:
5814:
5803:
5793:, we can obtain
5792:
5783:
5781:
5772:
5763:
5756:
5748:
5744:
5734:
5705:
5695:
5689:
5685:
5675:
5669:
5665:
5659:
5654:acting on a set
5653:
5647:
5630:
5624:
5622:
5621:
5616:
5611:
5606:
5605:
5596:
5591:
5586:
5578:
5567:
5566:
5543:
5529:
5513:
5500:
5481:
5471:
5457:between the set
5452:
5446:
5436:
5430:
5424:
5410:
5396:lie in the same
5395:
5389:
5381:
5363:In other words,
5362:
5360:
5359:
5354:
5349:
5348:
5325:
5324:
5309:
5308:
5282:
5274:
5273:
5253:
5239:
5192:
5182:
5171:
5157:
5143:
5137:
5127:
5126:
5114:
5106:
5100:
5094:
5083:
5075:
5069:
5063:
5055:
5038:
5024:
5009:
4991:
4973:
4967:
4945:
4930:
4907:
4896:
4885:
4871:
4865:
4859:
4853:
4844:
4838:
4832:
4826:
4811:
4800:
4790:
4784:
4778:
4768:
4766:
4765:
4760:
4743:
4717:
4716:
4700:
4694:
4680:
4675:with respect to
4674:
4667:
4666:
4660:
4654:
4648:
4642:
4636:
4630:
4624:
4610:
4604:
4598:
4592:
4578:
4567:derived functors
4564:
4558:
4548:
4540:
4533:
4527:
4518:
4512:
4506:
4500:
4490:
4476:
4466:
4457:
4444:
4438:
4432:
4422:
4416:
4407:
4401:
4395:
4389:
4383:
4377:
4371:
4357:
4348:
4342:
4332:
4326:
4320:
4306:
4292:
4286:invariant under
4283:
4277:
4250:denotes the set
4249:
4239:
4229:
4210:group cohomology
4199:
4193:
4181:
4180:
4173:
4172:
4165:
4164:
4158:
4148:
4138:
4132:
4123:
4117:
4111:
4102:
4088:
4074:
4068:
4055:
4037:
4031:
4021:
4007:
4001:
3992:
3978:
3968:
3962:
3956:
3948:
3946:
3945:
3940:
3917:
3900:
3885:
3875:
3869:
3863:
3857:
3851:
3845:
3838:
3837:
3831:
3826:acting on a set
3825:
3815:
3809:
3803:
3793:
3784:
3752:
3741:commutative ring
3734:
3706:
3692:
3674:
3664:
3655:
3645:
3624:
3604:
3594:
3580:
3569:
3552:
3546:
3531:
3521:
3506:
3482:
3463:
3462:
3452:product topology
3449:
3431:
3421:
3401:
3388:
3374:
3358:
3349:
3339:free regular set
3332:
3312:
3297:
3291:
3266:
3243:
3242:∖ {(0, 0)}
3236:
3230:
3215:
3205:
3178:
3174:
3168:
3161:
3151:
3136:
3126:
3116:
3110:
3097:
3082:
3072:
3063:
3042:
3032:
3010:
3004:
2990:
2984:
2959:orthogonal group
2956:
2946:
2939:
2926:
2918:
2906:
2901:has cardinality
2900:
2894:
2888:
2882:
2876:
2862:
2856:
2855:
2852:
2833:
2826:
2820:
2810:
2786:
2776:
2766:
2746:
2726:
2687:
2681:
2675:
2669:
2668:
2665:
2659:, the action is
2658:
2648:
2642:
2632:
2626:
2620:
2614:
2599:
2598:
2587:
2586:
2577:
2563:
2553:
2538:
2537:
2531:
2525:
2511:
2501:
2486:
2476:
2472:
2459:
2452:
2449:(of cardinality
2448:
2430:Cayley's theorem
2424:
2418:
2412:
2397:
2387:
2372:fixed-point free
2364:
2363:
2350:
2344:
2334:
2319:
2309:
2294:
2293:
2286:
2285:
2279:
2273:
2252:
2248:
2239:
2235:
2231:
2217:
2213:
2209:
2205:
2201:
2197:
2188:
2184:
2180:
2176:
2172:
2161:
2159:
2158:
2153:
2139:
2125:
2114:
2090:
2088:
2087:
2082:
2071:
2052:
2051:
2046:
2036:
2030:
2008:
2006:
2005:
2000:
1928:
1926:
1925:
1920:
1883:
1882:
1874:
1872:
1871:
1866:
1830:
1826:
1810:
1806:
1798:
1794:
1790:
1786:
1776:
1766:
1762:
1758:
1754:
1747:
1745:
1744:
1739:
1734:
1708:
1697:
1681:
1679:
1678:
1673:
1662:
1644:
1638:
1628:
1613:
1604:
1567:
1560:
1558:
1557:
1552:
1541:
1540:
1513:
1512:
1500:
1499:
1472:
1471:
1459:
1458:
1436:
1434:
1433:
1428:
1411:
1410:
1391:
1381:
1370:
1348:
1331:
1323:
1319:
1315:
1311:
1304:
1300:
1296:
1292:
1288:
1277:
1275:
1274:
1269:
1197:
1195:
1194:
1189:
1152:
1151:
1139:
1137:
1136:
1131:
1092:
1088:
1084:
1073:
1069:
1065:identity element
1058:
1033:
1023:
1006:
997:
984:
919:
909:
899:
890:
832:
809:
802:
795:
751:Algebraic groups
524:Hyperbolic group
514:Arithmetic group
505:
503:
502:
497:
495:
480:
478:
477:
472:
470:
443:
441:
440:
435:
433:
356:Schur multiplier
310:Cauchy's theorem
298:Quaternion group
246:
72:
61:
48:
37:
21:
8786:
8785:
8781:
8780:
8779:
8777:
8776:
8775:
8746:
8745:
8727:
8726:
8708:
8705:
8688:
8665:
8649:
8642:
8640:
8628:
8623:
8602:
8589:Kleinian groups
8586:
8573:
8560:
8554:
8541:
8535:
8518:
8504:
8486:
8480:
8467:
8461:
8448:
8434:
8418:
8415:
8410:
8409:
8401:
8397:
8389:
8385:
8377:
8373:
8366:
8353:
8352:
8348:
8336:
8335:
8331:
8322:
8318:
8308:
8306:
8304:
8289:
8288:
8284:
8276:
8272:
8264:
8260:
8252:
8248:
8240:
8236:
8228:
8224:
8216:
8212:
8202:
8200:
8192:
8191:
8187:
8175:
8173:
8169:
8157:
8156:
8152:
8147:
8142:
8110:
8103:
8100:
8091:
8088:
8079:
8037:In addition to
8021:
8011:
8001:
7987:
7974:
7968:
7958:
7952:
7938:
7911:
7901:
7895:
7889:
7883:
7877:
7871:
7858:
7852:
7842:
7836:
7830:
7820:
7814:
7805:
7799:
7793:
7780:
7770:
7764:
7758:
7746:
7740:
7727:
7715:
7714:. Morphisms of
7709:
7703:
7697:
7691:
7664:
7650:
7644:
7638:
7628:
7622:
7616:
7613:
7590:
7581:
7575:
7563:
7549:
7543:
7537:
7530:
7522:
7505:
7499:
7481:
7475:
7469:
7463:
7457:
7451:
7445:
7438:
7434:
7424:
7418:
7412:
7392:
7373:
7367:
7338:
7332:
7326:
7320:
7293:
7287:
7277:
7271:
7265:
7247:
7233:
7215:
7201:
7190:
7176:
7161:
7145:
7137:
7128:
7127:seconds ago if
7121:
7120:is positive or
7115:
7109:
7099:
7093:
7087:
7081:
7075:
7047:
7035:
7029:
7023:
7017:
7005:
7004:. Subgroups of
6999:
6993:
6983:
6981:field extension
6967:
6961:
6934:
6920:
6914:
6904:
6898:
6888:
6866:
6851:
6840:
6825:
6790:
6780:
6768:
6759:
6753:
6737:
6722:
6710:
6695:
6679:
6661:
6652:
6645:
6638:
6629:
6622:
6612:
6598:
6591:
6585:
6579:
6555:
6552:
6546:
6537:
6527:
6517:
6511:
6501:
6495:
6486:
6480:
6474:
6468:
6454:
6448:
6442:
6436:
6430:
6429:In every group
6417:
6404:
6390:
6384:
6378:
6368:
6367:In every group
6355:
6349:
6339:
6333:
6327:
6321:
6307:
6297:
6291:
6285:
6279:
6278:In every group
6270:
6260:
6254:
6248:
6234:
6228:
6222:
6216:
6215:In every group
6207:
6197:
6191:
6185:
6179:
6165:
6159:
6153:
6147:
6146:
6141:
6122:
6112:
6106:
6105:Fixing a group
6097:
6091:
6085:
6079:
6050:
6012:
5978:
5977:
5963:
5961:
5955:
5952:
5948:
5944:
5937:
5931:
5927:
5920:
5918:
5912:
5907:
5903:
5899:
5892:
5888:
5884:
5877:
5876:| = |
5874:
5870:
5863:
5861:
5855:
5848:
5846:
5838:
5832:
5827:
5823:
5816:
5812:
5805:
5804:| = |
5802:
5796:
5794:
5791:
5785:
5780:
5774:
5771:
5765:
5758:
5757:| = |
5752:
5750:
5746:
5740:
5730:
5725:. Consider the
5701:
5691:
5687:
5677:
5671:
5667:
5661:
5655:
5649:
5643:
5626:
5597:
5558:
5520:
5519:
5509:
5491:
5483:
5473:
5470:
5458:
5448:
5438:
5432:
5426:
5415:
5409:
5401:
5391:
5385:
5364:
5340:
5316:
5297:
5262:
5195:
5194:
5184:
5173:
5159:
5145:
5139:
5133:
5130:
5124:
5123:
5108:
5102:
5096:
5093:
5085:
5077:
5071:
5065:
5057:
5051:
5048:conjugacy class
5026:
5023:
5011:
5008:
4993:
4975:
4969:
4947:
4946:if and only if
4944:
4932:
4926:
4917:
4909:
4908:are related by
4906:
4898:
4895:
4887:
4873:
4867:
4861:
4855:
4849:
4840:
4834:
4828:
4825:
4817:
4802:
4796:
4786:
4780:
4774:
4708:
4703:
4702:
4696:
4690:
4676:
4670:
4664:
4663:
4656:
4650:
4644:
4638:
4632:
4626:
4612:
4606:
4600:
4594:
4588:
4585:
4574:
4560:
4554:
4544:
4536:
4529:
4523:
4514:
4513:and called the
4508:
4502:
4492:
4478:
4468:
4462:
4453:
4440:
4434:
4428:
4418:
4412:
4403:
4397:
4391:
4385:
4379:
4373:
4359:
4353:
4344:
4338:
4328:
4322:
4308:
4294:
4288:
4279:
4251:
4241:
4235:
4225:
4222:
4195:
4192:
4184:
4178:
4177:
4170:
4169:
4162:
4161:
4150:
4140:
4134:
4128:
4124:is non-empty).
4119:
4113:
4107:
4090:
4076:
4070:
4064:
4039:
4033:
4027:
4009:
4003:
3997:
3996:there exists a
3984:
3974:
3964:
3958:
3952:
3888:
3887:
3877:
3871:
3870:. The orbit of
3865:
3859:
3853:
3847:
3841:
3835:
3834:
3827:
3821:
3811:
3805:
3795:
3789:
3780:
3769:
3748:
3730:
3727:
3721:
3694:
3684:
3670:
3660:
3647:
3633:
3622:
3606:
3596:
3582:
3579:
3571:
3565:
3548:
3533:
3523:
3512:
3484:
3466:
3460:
3459:
3437:
3427:
3417:
3414:
3408:
3393:
3376:
3366:
3354:
3345:
3330:
3314:
3299:
3293:
3283:
3245:
3238:
3232:
3217:
3207:
3197:
3176:
3170:
3163:
3157:
3138:
3128:
3118:
3112:
3102:
3084:
3074:
3068:
3055:
3038:
3028:
3025:
3006:
3000:
2995:if there is no
2986:
2980:
2977:
2971:
2952:
2941:
2935:
2920:
2912:
2902:
2896:
2890:
2884:
2878:
2872:
2869:
2858:
2850:
2847:
2846:
2828:
2822:
2812:
2809:
2800:
2788:
2778:
2768:
2765:
2756:
2748:
2745:
2736:
2728:
2721:
2712:
2705:
2696:
2689:
2683:
2677:
2671:
2663:
2661:
2660:
2653:
2652:For an integer
2644:
2638:
2628:
2622:
2616:
2602:
2596:
2595:
2584:
2583:
2565:
2555:
2554:there exists a
2541:
2535:
2534:
2527:
2521:
2518:
2503:
2492:
2488:
2485:
2481:
2474:
2464:
2454:
2450:
2438:
2420:
2414:
2411:
2399:
2389:
2375:
2361:
2360:
2346:
2340:
2333:
2321:
2311:
2297:
2291:
2290:
2283:
2282:
2275:
2269:
2266:
2250:
2244:
2237:
2233:
2219:
2215:
2211:
2207:
2203:
2199:
2193:
2186:
2182:
2178:
2174:
2170:
2099:
2098:
2095:Compatibility:
2059:
2058:
2038:
2032:
2017:
1937:
1936:
1933:Compatibility:
1890:
1889:
1836:
1835:
1828:
1824:
1817:
1808:
1800:
1796:
1792:
1788:
1782:
1768:
1764:
1760:
1756:
1752:
1685:
1684:
1650:
1649:
1640:
1630:
1615:
1609:
1603:
1594:
1585:
1577:
1565:
1529:
1504:
1491:
1463:
1450:
1445:
1444:
1402:
1397:
1396:
1383:
1380:
1372:
1361:
1353:
1351:transformations
1344:
1329:
1321:
1317:
1313:
1309:
1302:
1298:
1294:
1290:
1286:
1206:
1205:
1202:Compatibility:
1159:
1158:
1101:
1100:
1090:
1086:
1082:
1071:
1067:
1056:
1053:
1048:
1029:
1022:
1014:
1012:symmetric group
1002:
993:
974:
929:Euclidean space
915:
905:
895:
886:
847:transformations
845:, many sets of
831:
827:
813:
784:
783:
772:Abelian variety
765:Reductive group
753:
743:
742:
741:
740:
691:
683:
675:
667:
659:
632:Special unitary
543:
529:
528:
510:
509:
486:
485:
461:
460:
424:
423:
415:
414:
405:Discrete groups
394:
393:
349:Frobenius group
294:
281:
270:
263:Symmetric group
259:
243:
233:
232:
83:Normal subgroup
69:
49:
40:
35:
32:
23:
22:
15:
12:
11:
5:
8784:
8782:
8774:
8773:
8768:
8763:
8758:
8748:
8747:
8744:
8743:
8732:"Group Action"
8724:
8704:
8703:External links
8701:
8700:
8699:
8686:
8663:
8647:
8626:
8621:
8600:
8584:
8571:
8558:
8552:
8539:
8533:
8516:
8502:
8488:Hatcher, Allen
8484:
8478:
8465:
8459:
8446:
8432:
8414:
8411:
8408:
8407:
8403:Perrone (2024)
8395:
8391:Perrone (2024)
8383:
8381:, pp. 7–9
8379:Perrone (2024)
8371:
8364:
8346:
8343:. p. 145.
8329:
8316:
8302:
8282:
8278:tom Dieck 1987
8270:
8258:
8246:
8244:, p. 176.
8234:
8222:
8210:
8185:
8182:. p. 253.
8176:Smith (2008).
8167:
8164:. p. 144.
8149:
8148:
8146:
8143:
8141:
8138:
8137:
8136:
8131:
8126:
8121:
8116:
8109:
8106:
8105:
8104:
8101:
8094:
8092:
8089:
8082:
8078:
8075:
8043:smooth actions
7937:
7934:
7916:-sets forms a
7908:
7907:
7868:
7811:
7792:Every regular
7757:, and the two
7649:is a function
7612:
7605:
7577:Main article:
7574:
7571:
7570:
7569:
7518:
7442:
7391:, the mapping
7352:
7262:
7134:
7041:
6973:
6909:-sets and the
6885:
6874:
6819:
6796:
6765:
6676:
6669:
6657:
6650:
6643:
6634:
6627:
6620:
6597:, the mapping
6573:
6570:
6567:symmetry group
6563:
6548:
6543:
6492:
6485:conjugates of
6435:with subgroup
6427:
6365:
6284:with subgroup
6276:
6213:
6164:is defined by
6140:
6137:
6129:disjoint union
6067:
6063:
6057:
6053:
6048:
6042:
6039:
6036:
6032:
6024:
6020:
6016:
6011:
6006:
6002:
5998:
5994:
5990:
5986:
5970:
5969:
5950:
5946:
5942:
5929:
5925:
5905:
5901:
5897:
5886:
5882:
5872:
5868:
5853:
5825:
5821:
5810:
5800:
5789:
5778:
5769:
5747:{1, 2, ..., 8}
5698:
5697:
5614:
5610:
5604:
5600:
5595:
5590:
5585:
5581:
5577:
5573:
5570:
5565:
5561:
5556:
5552:
5549:
5546:
5542:
5538:
5535:
5532:
5528:
5487:
5482:, which sends
5466:
5456:
5405:
5383:if and only if
5352:
5347:
5343:
5339:
5336:
5333:
5329:
5323:
5319:
5315:
5312:
5307:
5304:
5300:
5295:
5291:
5288:
5285:
5281:
5277:
5272:
5269:
5265:
5260:
5256:
5252:
5248:
5245:
5242:
5238:
5234:
5230:
5226:
5223:
5220:
5217:
5214:
5211:
5208:
5205:
5202:
5129:
5121:
5089:
5019:
5004:
4940:
4922:
4913:
4902:
4891:
4821:
4758:
4755:
4752:
4749:
4746:
4742:
4738:
4735:
4732:
4729:
4726:
4723:
4720:
4715:
4711:
4683:isotropy group
4584:
4581:
4549:is the zeroth
4337:the action to
4284:is said to be
4221:
4218:
4214:group homology
4207:
4203:
4188:
4183:, and written
4139:is written as
4106:
3994:if and only if
3938:
3935:
3932:
3929:
3926:
3923:
3920:
3916:
3912:
3909:
3906:
3903:
3899:
3895:
3876:is denoted by
3840:of an element
3768:
3765:
3723:Main article:
3720:
3719:Linear actions
3717:
3618:
3575:
3555:discrete group
3410:Main article:
3407:
3404:
3391:quotient space
3326:
3280:covering space
3270:The action by
3188:The action is
3050:The action is
3045:homeomorphisms
3024:
3021:
2979:The action of
2973:Main article:
2970:
2967:
2927:-transitive.
2868:
2865:
2805:
2796:
2761:
2752:
2741:
2732:
2717:
2710:
2701:
2694:
2581:The action is
2520:The action of
2517:
2514:
2490:
2483:
2407:
2329:
2265:
2262:
2242:opposite group
2167:
2166:
2163:
2162:
2151:
2148:
2145:
2142:
2138:
2134:
2131:
2128:
2124:
2120:
2117:
2113:
2109:
2106:
2096:
2092:
2091:
2080:
2077:
2074:
2070:
2066:
2056:
2014:
2013:
2010:
2009:
1998:
1995:
1992:
1989:
1986:
1983:
1980:
1977:
1974:
1971:
1968:
1965:
1962:
1959:
1956:
1953:
1950:
1947:
1944:
1934:
1930:
1929:
1918:
1915:
1912:
1909:
1906:
1903:
1900:
1897:
1887:
1876:
1875:
1864:
1861:
1858:
1855:
1852:
1849:
1846:
1843:
1831:is a function
1816:
1813:
1749:
1748:
1737:
1733:
1729:
1726:
1723:
1720:
1717:
1714:
1711:
1707:
1703:
1700:
1696:
1692:
1682:
1671:
1668:
1665:
1661:
1657:
1599:
1590:
1581:
1562:
1561:
1550:
1547:
1544:
1539:
1536:
1532:
1528:
1525:
1522:
1519:
1516:
1511:
1507:
1503:
1498:
1494:
1490:
1487:
1484:
1481:
1478:
1475:
1470:
1466:
1462:
1457:
1453:
1438:
1437:
1426:
1423:
1420:
1417:
1414:
1409:
1405:
1376:
1357:
1283:
1282:
1279:
1278:
1267:
1264:
1261:
1258:
1255:
1252:
1249:
1246:
1243:
1240:
1237:
1234:
1231:
1228:
1225:
1222:
1219:
1216:
1213:
1203:
1199:
1198:
1187:
1184:
1181:
1178:
1175:
1172:
1169:
1166:
1156:
1141:
1140:
1129:
1126:
1123:
1120:
1117:
1114:
1111:
1108:
1052:
1049:
1047:
1044:
1018:
964:representation
863:abstract group
829:
815:
814:
812:
811:
804:
797:
789:
786:
785:
782:
781:
779:Elliptic curve
775:
774:
768:
767:
761:
760:
754:
749:
748:
745:
744:
739:
738:
735:
732:
728:
724:
723:
722:
717:
715:Diffeomorphism
711:
710:
705:
700:
694:
693:
689:
685:
681:
677:
673:
669:
665:
661:
657:
652:
651:
640:
639:
628:
627:
616:
615:
604:
603:
592:
591:
580:
579:
572:Special linear
568:
567:
560:General linear
556:
555:
550:
544:
535:
534:
531:
530:
527:
526:
521:
516:
508:
507:
494:
482:
469:
456:
454:Modular groups
452:
451:
450:
445:
432:
416:
413:
412:
407:
401:
400:
399:
396:
395:
390:
389:
388:
387:
382:
377:
374:
368:
367:
361:
360:
359:
358:
352:
351:
345:
344:
339:
330:
329:
327:Hall's theorem
324:
322:Sylow theorems
318:
317:
312:
304:
303:
302:
301:
295:
290:
287:Dihedral group
283:
282:
277:
271:
266:
260:
255:
244:
239:
238:
235:
234:
229:
228:
227:
226:
221:
213:
212:
211:
210:
205:
200:
195:
190:
185:
180:
178:multiplicative
175:
170:
165:
160:
152:
151:
150:
149:
144:
136:
135:
127:
126:
125:
124:
122:Wreath product
119:
114:
109:
107:direct product
101:
99:Quotient group
93:
92:
91:
90:
85:
80:
70:
67:
66:
63:
62:
54:
53:
33:
24:
14:
13:
10:
9:
6:
4:
3:
2:
8783:
8772:
8769:
8767:
8764:
8762:
8759:
8757:
8754:
8753:
8751:
8739:
8738:
8733:
8730:
8725:
8721:
8717:
8716:
8711:
8707:
8706:
8702:
8697:
8693:
8689:
8683:
8679:
8675:
8671:
8670:
8664:
8661:
8657:
8653:
8648:
8639:on 2020-07-27
8638:
8634:
8633:
8627:
8624:
8618:
8614:
8610:
8606:
8601:
8598:
8594:
8590:
8585:
8582:
8578:
8574:
8568:
8564:
8559:
8555:
8549:
8545:
8540:
8536:
8534:0-387-94285-8
8530:
8526:
8522:
8517:
8513:
8509:
8505:
8499:
8495:
8494:
8489:
8485:
8481:
8475:
8471:
8466:
8462:
8460:0-471-43334-9
8456:
8452:
8447:
8443:
8439:
8435:
8429:
8425:
8421:
8417:
8416:
8412:
8404:
8399:
8396:
8392:
8387:
8384:
8380:
8375:
8372:
8367:
8365:9780521613255
8361:
8357:
8350:
8347:
8342:
8341:
8333:
8330:
8326:
8320:
8317:
8305:
8303:9780387289298
8299:
8295:
8294:
8286:
8283:
8279:
8274:
8271:
8267:
8262:
8259:
8255:
8250:
8247:
8243:
8242:Thurston 1980
8238:
8235:
8231:
8230:Kapovich 2009
8226:
8223:
8219:
8218:Thurston 1997
8214:
8211:
8199:
8195:
8189:
8186:
8181:
8180:
8171:
8168:
8163:
8162:
8154:
8151:
8144:
8139:
8135:
8132:
8130:
8129:Monoid action
8127:
8125:
8122:
8120:
8117:
8115:
8112:
8111:
8107:
8098:
8093:
8086:
8081:
8076:
8074:
8072:
8071:group objects
8068:
8064:
8063:group schemes
8060:
8056:
8052:
8048:
8044:
8040:
8035:
8033:
8029:
8024:
8019:
8014:
8009:
8004:
7999:
7995:
7990:
7984:
7982:
7977:
7971:
7966:
7965:endomorphisms
7961:
7955:
7949:
7947:
7943:
7935:
7933:
7931:
7927:
7923:
7919:
7914:
7904:
7898:
7892:
7886:
7880:
7874:
7869:
7865:
7861:
7855:
7849:
7845:
7839:
7833:
7827:
7823:
7817:
7812:
7808:
7802:
7796:
7791:
7790:
7789:
7786:
7783:
7778:
7773:
7767:
7761:
7756:
7755:
7751:is called an
7749:
7743:
7737:
7735:
7730:
7725:
7724:
7718:
7712:
7706:
7700:
7694:
7687:
7683:
7679:
7675:
7671:
7667:
7661:
7657:
7653:
7647:
7641:
7636:
7631:
7625:
7619:
7610:
7606:
7604:
7601:
7597:
7593:
7589:
7588:
7580:
7572:
7566:
7561:
7556:
7552:
7546:
7540:
7534:
7526:
7521:
7516:
7512:
7508:
7502:
7496:
7492:
7488:
7484:
7478:
7472:
7466:
7460:
7454:
7448:
7443:
7432:
7427:
7421:
7415:
7409:
7406:
7403:
7399:
7395:
7388:
7384:
7380:
7376:
7370:
7365:
7361:
7357:
7353:
7350:
7346:
7345:Mathieu group
7341:
7335:
7329:
7323:
7316:
7312:
7308:
7304:
7300:
7296:
7292:, by setting
7290:
7285:
7280:
7274:
7268:
7263:
7258:
7254:
7250:
7244:
7240:
7236:
7230:
7226:
7222:
7218:
7212:
7208:
7204:
7197:
7193:
7187:
7183:
7179:
7172:
7168:
7164:
7157:
7153:
7149:
7141:
7135:
7131:
7125:
7118:
7112:
7106:
7102:
7096:
7090:
7084:
7078:
7073:
7069:
7065:
7062:" systems in
7061:
7057:
7051:
7046:
7042:
7038:
7032:
7026:
7022:that contain
7020:
7013:
7009:
7002:
6996:
6990:
6986:
6982:
6978:
6974:
6970:
6964:
6957:
6953:
6949:
6945:
6941:
6937:
6931:
6927:
6923:
6917:
6912:
6907:
6901:
6896:
6893:comprise the
6891:
6886:
6883:
6879:
6875:
6870:
6865:
6861:
6855:
6847:
6843:
6839:
6833:
6829:
6824:
6820:
6817:
6813:
6809:
6805:
6801:
6797:
6793:
6788:
6783:
6776:
6772:
6766:
6762:
6756:
6751:
6745:
6741:
6736:
6730:
6726:
6718:
6714:
6709:
6703:
6699:
6693:
6692:Lie subgroups
6687:
6683:
6677:
6674:
6670:
6667:
6660:
6656:
6649:
6642:
6637:
6633:
6626:
6619:
6615:
6609:
6605:
6601:
6594:
6588:
6584:over a field
6582:
6578:
6574:
6571:
6568:
6564:
6559:
6551:
6544:
6540:
6535:
6530:
6524:
6520:
6514:
6509:
6504:
6498:
6494:An action of
6493:
6489:
6483:
6477:
6471:
6465:
6461:
6457:
6451:
6445:
6439:
6433:
6428:
6424:
6420:
6414:
6411:
6407:
6401:
6397:
6393:
6387:
6381:
6376:
6371:
6366:
6363:
6358:
6352:
6347:
6342:
6336:
6330:
6324:
6318:
6314:
6310:
6304:
6300:
6294:
6288:
6282:
6277:
6273:
6268:
6263:
6257:
6251:
6245:
6241:
6237:
6231:
6225:
6219:
6214:
6210:
6205:
6200:
6194:
6188:
6182:
6176:
6172:
6168:
6162:
6156:
6151:
6143:
6142:
6138:
6136:
6134:
6130:
6125:
6120:
6119:Burnside ring
6115:
6109:
6103:
6100:
6094:
6088:
6082:
6065:
6055:
6051:
6040:
6037:
6034:
6030:
6018:
6009:
6004:
5996:
5992:
5988:
5975:
5958:
5941:
5924:
5915:
5896:
5891:| |
5881:
5867:
5852:
5842:
5835:
5820:
5815:| |
5809:
5799:
5788:
5777:
5768:
5764:| |
5761:
5755:
5743:
5738:
5733:
5728:
5727:cubical graph
5724:
5720:
5717:
5716:
5711:
5707:
5704:
5694:
5684:
5680:
5674:
5664:
5658:
5652:
5646:
5641:
5638:
5637:
5636:
5634:
5629:
5612:
5602:
5598:
5588:
5579:
5571:
5563:
5559:
5554:
5550:
5544:
5536:
5533:
5530:
5517:
5512:
5506:
5504:
5499:
5495:
5490:
5486:
5480:
5476:
5469:
5465:
5461:
5454:
5451:
5445:
5441:
5435:
5429:
5422:
5418:
5414:
5408:
5404:
5399:
5394:
5388:
5384:
5379:
5375:
5371:
5367:
5350:
5345:
5341:
5337:
5334:
5331:
5321:
5317:
5313:
5310:
5305:
5302:
5298:
5289:
5286:
5283:
5279:
5275:
5270:
5267:
5263:
5254:
5250:
5246:
5243:
5240:
5236:
5232:
5221:
5215:
5212:
5206:
5200:
5191:
5187:
5180:
5176:
5170:
5166:
5162:
5156:
5152:
5148:
5142:
5136:
5120:
5118:
5112:
5105:
5099:
5092:
5088:
5081:
5074:
5068:
5061:
5054:
5049:
5045:
5040:
5037:
5033:
5029:
5022:
5018:
5014:
5007:
5003:
4999:
4996:
4990:
4986:
4982:
4979:
4972:
4966:
4962:
4958:
4954:
4950:
4943:
4939:
4935:
4929:
4925:
4921:
4916:
4912:
4905:
4901:
4894:
4890:
4884:
4880:
4876:
4870:
4864:
4858:
4852:
4846:
4843:
4837:
4831:
4824:
4820:
4815:
4809:
4805:
4799:
4794:
4789:
4783:
4777:
4772:
4756:
4750:
4747:
4744:
4740:
4736:
4733:
4730:
4727:
4724:
4718:
4713:
4709:
4699:
4693:
4688:
4684:
4679:
4673:
4668:
4659:
4653:
4649:". For every
4647:
4641:
4635:
4629:
4623:
4619:
4615:
4609:
4603:
4597:
4591:
4582:
4580:
4579:-invariants.
4577:
4572:
4568:
4563:
4557:
4552:
4547:
4542:
4539:
4532:
4526:
4521:
4517:
4511:
4505:
4499:
4495:
4489:
4485:
4481:
4475:
4471:
4465:
4460:
4456:
4450:
4448:
4443:
4437:
4431:
4426:
4421:
4415:
4409:
4406:
4400:
4394:
4388:
4382:
4376:
4370:
4366:
4362:
4356:
4352:
4347:
4343:. The subset
4341:
4336:
4331:
4325:
4319:
4315:
4311:
4305:
4301:
4297:
4291:
4287:
4282:
4278:. The subset
4275:
4271:
4267:
4263:
4259:
4255:
4248:
4244:
4238:
4233:
4228:
4219:
4217:
4215:
4211:
4205:
4201:
4198:
4191:
4187:
4182:
4174:
4166:
4157:
4153:
4147:
4143:
4137:
4131:
4125:
4122:
4116:
4110:
4104:
4101:
4097:
4093:
4087:
4083:
4079:
4073:
4067:
4062:
4057:
4054:
4050:
4046:
4042:
4036:
4030:
4025:
4020:
4016:
4012:
4006:
4000:
3995:
3991:
3987:
3982:
3977:
3972:
3967:
3961:
3955:
3949:
3936:
3930:
3927:
3924:
3921:
3918:
3914:
3910:
3904:
3901:
3897:
3893:
3884:
3880:
3874:
3868:
3862:
3856:
3850:
3844:
3839:
3830:
3824:
3814:
3808:
3802:
3798:
3792:
3788:
3783:
3778:
3773:
3766:
3764:
3762:
3758:
3757:
3751:
3746:
3742:
3738:
3733:
3726:
3718:
3716:
3714:
3710:
3705:
3701:
3697:
3691:
3687:
3682:
3681:smooth points
3678:
3673:
3668:
3663:
3657:
3654:
3650:
3644:
3640:
3636:
3631:
3626:
3621:
3617:
3613:
3609:
3603:
3599:
3593:
3589:
3585:
3578:
3574:
3568:
3563:
3558:
3556:
3551:
3544:
3540:
3536:
3530:
3526:
3519:
3515:
3510:
3504:
3500:
3496:
3492:
3488:
3481:
3477:
3473:
3469:
3464:
3455:
3453:
3448:
3444:
3440:
3435:
3430:
3425:
3420:
3413:
3405:
3403:
3400:
3396:
3392:
3387:
3383:
3379:
3373:
3369:
3364:
3363:
3357:
3353:
3348:
3342:
3340:
3336:
3329:
3325:
3321:
3317:
3310:
3306:
3302:
3296:
3290:
3286:
3281:
3277:
3273:
3268:
3264:
3260:
3256:
3252:
3248:
3241:
3235:
3228:
3224:
3220:
3214:
3210:
3204:
3200:
3195:
3192:if for every
3191:
3186:
3184:
3183:
3182:wandering set
3173:
3167:
3160:
3155:
3149:
3145:
3141:
3135:
3131:
3125:
3121:
3115:
3109:
3105:
3099:
3095:
3091:
3087:
3081:
3077:
3071:
3067:
3066:neighbourhood
3062:
3058:
3053:
3048:
3046:
3041:
3036:
3031:
3022:
3020:
3018:
3014:
3009:
3003:
2998:
2994:
2989:
2983:
2976:
2968:
2966:
2964:
2960:
2955:
2950:
2944:
2938:
2933:
2928:
2924:
2916:
2910:
2905:
2899:
2893:
2887:
2881:
2875:
2866:
2864:
2861:
2854:
2845:An action is
2843:
2841:
2837:
2831:
2825:
2819:
2815:
2808:
2804:
2799:
2795:
2791:
2785:
2781:
2775:
2771:
2764:
2760:
2755:
2751:
2744:
2740:
2735:
2731:
2725:
2720:
2716:
2709:
2704:
2700:
2693:
2686:
2680:
2676:has at least
2674:
2667:
2656:
2650:
2647:
2641:
2636:
2631:
2625:
2619:
2613:
2609:
2605:
2600:
2592:
2588:
2579:
2576:
2572:
2568:
2562:
2558:
2552:
2548:
2544:
2539:
2530:
2524:
2515:
2513:
2510:
2506:
2500:
2496:
2478:
2471:
2467:
2463:
2458:
2446:
2442:
2435:
2431:
2426:
2423:
2417:
2410:
2406:
2402:
2396:
2392:
2386:
2382:
2378:
2373:
2369:
2365:
2356:
2354:
2349:
2343:
2338:
2332:
2328:
2324:
2320:implies that
2318:
2314:
2308:
2304:
2300:
2295:
2287:
2278:
2272:
2263:
2261:
2259:
2254:
2247:
2243:
2230:
2227:
2223:
2196:
2190:
2146:
2143:
2136:
2132:
2129:
2126:
2122:
2115:
2111:
2107:
2097:
2094:
2093:
2078:
2075:
2072:
2068:
2064:
2057:
2054:
2053:
2050:
2049:
2048:
2045:
2041:
2035:
2028:
2024:
2020:
1993:
1990:
1987:
1984:
1978:
1975:
1969:
1966:
1960:
1957:
1954:
1948:
1942:
1935:
1932:
1931:
1916:
1913:
1907:
1904:
1901:
1895:
1888:
1885:
1884:
1881:
1880:
1879:
1862:
1859:
1853:
1850:
1847:
1844:
1841:
1834:
1833:
1832:
1822:
1814:
1812:
1804:
1785:
1780:
1775:
1771:
1735:
1731:
1724:
1721:
1715:
1709:
1705:
1701:
1694:
1690:
1683:
1669:
1666:
1663:
1659:
1655:
1648:
1647:
1646:
1643:
1637:
1633:
1626:
1622:
1618:
1612:
1606:
1602:
1598:
1593:
1589:
1584:
1580:
1575:
1571:
1545:
1537:
1534:
1530:
1526:
1520:
1509:
1505:
1501:
1496:
1492:
1485:
1476:
1468:
1464:
1455:
1451:
1443:
1442:
1441:
1424:
1421:
1415:
1407:
1403:
1395:
1394:
1393:
1390:
1386:
1379:
1375:
1369:
1365:
1360:
1356:
1352:
1347:
1342:
1337:
1335:
1327:
1324:is called a (
1306:
1262:
1259:
1256:
1253:
1247:
1244:
1235:
1232:
1229:
1223:
1220:
1217:
1211:
1204:
1201:
1200:
1185:
1182:
1176:
1173:
1170:
1164:
1157:
1154:
1153:
1150:
1149:
1148:
1146:
1127:
1124:
1118:
1115:
1112:
1109:
1106:
1099:
1098:
1097:
1096:
1081:
1077:
1066:
1062:
1050:
1045:
1043:
1041:
1037:
1032:
1027:
1021:
1017:
1013:
1008:
1005:
1001:
996:
992:
988:
982:
978:
973:
969:
965:
961:
956:
954:
950:
946:
942:
938:
934:
930:
926:
921:
918:
913:
908:
903:
898:
894:
889:
884:
879:
877:
873:
868:
864:
860:
856:
852:
848:
844:
836:
826:
821:
810:
805:
803:
798:
796:
791:
790:
788:
787:
780:
777:
776:
773:
770:
769:
766:
763:
762:
759:
756:
755:
752:
747:
746:
736:
733:
730:
729:
727:
721:
718:
716:
713:
712:
709:
706:
704:
701:
699:
696:
695:
692:
686:
684:
678:
676:
670:
668:
662:
660:
654:
653:
649:
645:
642:
641:
637:
633:
630:
629:
625:
621:
618:
617:
613:
609:
606:
605:
601:
597:
594:
593:
589:
585:
582:
581:
577:
573:
570:
569:
565:
561:
558:
557:
554:
551:
549:
546:
545:
542:
538:
533:
532:
525:
522:
520:
517:
515:
512:
511:
483:
458:
457:
455:
449:
446:
421:
418:
417:
411:
408:
406:
403:
402:
398:
397:
386:
383:
381:
378:
375:
372:
371:
370:
369:
366:
362:
357:
354:
353:
350:
347:
346:
343:
340:
338:
336:
332:
331:
328:
325:
323:
320:
319:
316:
313:
311:
308:
307:
306:
305:
299:
296:
293:
288:
285:
284:
280:
275:
272:
269:
264:
261:
258:
253:
250:
249:
248:
247:
242:
241:Finite groups
237:
236:
225:
222:
220:
217:
216:
215:
214:
209:
206:
204:
201:
199:
196:
194:
191:
189:
186:
184:
181:
179:
176:
174:
171:
169:
166:
164:
161:
159:
156:
155:
154:
153:
148:
145:
143:
140:
139:
138:
137:
134:
133:
128:
123:
120:
118:
115:
113:
110:
108:
105:
102:
100:
97:
96:
95:
94:
89:
86:
84:
81:
79:
76:
75:
74:
73:
68:Basic notions
65:
64:
60:
56:
55:
52:
47:
43:
38:
30:
19:
8756:Group theory
8735:
8713:
8668:
8651:
8641:, retrieved
8637:the original
8631:
8604:
8588:
8562:
8543:
8524:
8520:
8492:
8469:
8450:
8423:
8398:
8386:
8374:
8355:
8349:
8339:
8332:
8324:
8319:
8307:. Retrieved
8292:
8285:
8273:
8261:
8254:Hatcher 2002
8249:
8237:
8225:
8213:
8201:. Retrieved
8197:
8188:
8178:
8170:
8160:
8153:
8036:
8022:
8012:
8002:
7988:
7985:
7975:
7969:
7959:
7953:
7950:
7939:
7912:
7909:
7902:
7896:
7890:
7884:
7878:
7872:
7863:
7859:
7853:
7847:
7843:
7837:
7831:
7825:
7821:
7815:
7806:
7800:
7794:
7787:
7781:
7776:
7771:
7765:
7759:
7752:
7747:
7741:
7738:
7733:
7728:
7721:
7716:
7710:
7704:
7698:
7692:
7685:
7681:
7677:
7673:
7669:
7665:
7659:
7655:
7651:
7645:
7639:
7634:
7629:
7623:
7617:
7614:
7608:
7599:
7595:
7591:
7584:
7582:
7564:
7554:
7550:
7544:
7538:
7532:
7524:
7519:
7514:
7510:
7506:
7500:
7494:
7490:
7486:
7482:
7476:
7470:
7464:
7458:
7452:
7446:
7425:
7419:
7413:
7407:
7404:
7401:
7397:
7393:
7386:
7382:
7378:
7374:
7368:
7339:
7333:
7327:
7321:
7314:
7310:
7306:
7302:
7298:
7294:
7288:
7278:
7272:
7266:
7256:
7252:
7248:
7242:
7238:
7234:
7228:
7224:
7220:
7216:
7210:
7206:
7202:
7195:
7191:
7185:
7181:
7177:
7170:
7166:
7162:
7155:
7151:
7147:
7139:
7133:is negative.
7129:
7123:
7116:
7110:
7104:
7100:
7094:
7088:
7082:
7076:
7060:well-behaved
7054:acts on the
7049:
7045:real numbers
7036:
7030:
7024:
7018:
7011:
7007:
7000:
6994:
6988:
6984:
6977:Galois group
6968:
6962:
6955:
6951:
6947:
6943:
6939:
6935:
6929:
6925:
6921:
6915:
6905:
6899:
6889:
6868:
6864:Möbius group
6853:
6845:
6841:
6831:
6827:
6816:affine space
6811:
6808:affine space
6804:transitively
6800:affine group
6791:
6781:
6774:
6770:
6760:
6754:
6743:
6739:
6728:
6724:
6716:
6712:
6701:
6697:
6685:
6681:
6658:
6654:
6647:
6640:
6635:
6631:
6624:
6617:
6613:
6607:
6603:
6599:
6592:
6586:
6580:
6557:
6549:
6538:
6528:
6522:
6518:
6512:
6508:automorphism
6502:
6496:
6487:
6481:
6475:
6469:
6463:
6459:
6455:
6449:
6443:
6437:
6431:
6422:
6418:
6412:
6409:
6405:
6399:
6395:
6391:
6385:
6379:
6369:
6356:
6350:
6340:
6334:
6328:
6322:
6316:
6312:
6308:
6302:
6298:
6292:
6286:
6280:
6271:
6261:
6255:
6249:
6243:
6239:
6235:
6229:
6223:
6217:
6208:
6198:
6192:
6186:
6180:
6174:
6170:
6166:
6160:
6154:
6145:
6123:
6113:
6107:
6104:
6098:
6092:
6086:
6080:
5971:
5956:
5939:
5922:
5913:
5894:
5879:
5865:
5850:
5840:
5833:
5818:
5807:
5797:
5786:
5775:
5766:
5759:
5753:
5741:
5739:group. Then
5737:automorphism
5731:
5718:
5702:
5699:
5692:
5682:
5678:
5672:
5662:
5656:
5650:
5644:
5639:
5627:
5510:
5507:
5502:
5497:
5493:
5488:
5484:
5478:
5474:
5467:
5463:
5459:
5449:
5443:
5439:
5433:
5427:
5420:
5416:
5411:. Thus, the
5406:
5402:
5392:
5386:
5382:
5377:
5373:
5369:
5365:
5189:
5185:
5178:
5174:
5168:
5164:
5160:
5154:
5150:
5146:
5140:
5134:
5131:
5110:
5103:
5097:
5095:of some/any
5090:
5086:
5079:
5072:
5066:
5059:
5052:
5041:
5035:
5031:
5027:
5020:
5016:
5012:
5005:
5001:
4997:
4994:
4988:
4984:
4980:
4977:
4970:
4964:
4960:
4956:
4952:
4948:
4941:
4937:
4933:
4927:
4923:
4919:
4914:
4910:
4903:
4899:
4892:
4888:
4882:
4878:
4874:
4868:
4862:
4856:
4850:
4847:
4841:
4835:
4829:
4822:
4818:
4814:intersection
4807:
4803:
4797:
4787:
4781:
4775:
4697:
4691:
4687:little group
4686:
4682:
4677:
4671:
4662:
4657:
4651:
4645:
4639:
4633:
4627:
4621:
4617:
4613:
4607:
4601:
4595:
4589:
4586:
4575:
4561:
4555:
4545:
4537:
4530:
4524:
4519:
4515:
4509:
4503:
4497:
4493:
4487:
4483:
4479:
4473:
4469:
4463:
4458:
4454:
4451:
4446:
4441:
4435:
4429:
4425:transitively
4419:
4413:
4410:
4404:
4398:
4392:
4386:
4380:
4374:
4368:
4364:
4360:
4354:
4351:fixed under
4350:
4345:
4339:
4329:
4323:
4317:
4313:
4309:
4303:
4299:
4295:
4289:
4285:
4280:
4273:
4269:
4265:
4261:
4257:
4253:
4246:
4242:
4236:
4226:
4223:
4196:
4189:
4185:
4179:coinvariants
4176:
4168:
4160:
4155:
4151:
4145:
4141:
4135:
4129:
4126:
4120:
4118:(given that
4114:
4108:
4099:
4095:
4091:
4085:
4081:
4077:
4071:
4065:
4058:
4052:
4048:
4044:
4040:
4034:
4028:
4018:
4014:
4010:
4004:
3998:
3989:
3985:
3975:
3965:
3959:
3953:
3950:
3882:
3878:
3872:
3866:
3860:
3854:
3848:
3842:
3833:
3828:
3822:
3819:
3812:
3806:
3800:
3796:
3790:
3781:
3754:
3749:
3731:
3728:
3703:
3699:
3695:
3689:
3685:
3680:
3671:
3661:
3658:
3652:
3648:
3642:
3638:
3634:
3629:
3627:
3619:
3615:
3611:
3607:
3601:
3597:
3591:
3587:
3583:
3576:
3572:
3566:
3562:locally free
3561:
3559:
3549:
3542:
3538:
3534:
3528:
3524:
3517:
3513:
3502:
3498:
3494:
3490:
3486:
3479:
3475:
3471:
3467:
3458:
3456:
3446:
3442:
3438:
3433:
3428:
3418:
3415:
3398:
3394:
3385:
3381:
3377:
3371:
3367:
3360:
3355:
3346:
3343:
3338:
3334:
3327:
3323:
3319:
3315:
3308:
3304:
3300:
3294:
3288:
3284:
3269:
3262:
3258:
3254:
3250:
3246:
3239:
3233:
3226:
3222:
3218:
3212:
3208:
3202:
3198:
3189:
3187:
3180:
3171:
3165:
3158:
3153:
3147:
3143:
3139:
3133:
3129:
3123:
3119:
3113:
3107:
3103:
3100:
3093:
3089:
3085:
3079:
3075:
3069:
3060:
3056:
3051:
3049:
3039:
3029:
3027:Assume that
3026:
3007:
3001:
2992:
2987:
2981:
2978:
2953:
2942:
2936:
2929:
2922:
2914:
2903:
2897:
2891:
2885:
2879:
2873:
2870:
2859:
2848:
2844:
2829:
2823:
2817:
2813:
2806:
2802:
2797:
2793:
2789:
2783:
2779:
2773:
2769:
2762:
2758:
2753:
2749:
2742:
2738:
2733:
2729:
2723:
2718:
2714:
2707:
2702:
2698:
2691:
2684:
2678:
2672:
2662:
2654:
2651:
2645:
2639:
2629:
2623:
2617:
2615:the element
2611:
2607:
2603:
2594:
2590:
2582:
2580:
2574:
2570:
2566:
2560:
2556:
2550:
2546:
2542:
2533:
2528:
2522:
2519:
2508:
2504:
2498:
2494:
2479:
2469:
2465:
2462:cyclic group
2456:
2444:
2440:
2427:
2421:
2415:
2408:
2404:
2400:
2394:
2390:
2384:
2380:
2376:
2371:
2367:
2359:
2357:
2347:
2341:
2337:homomorphism
2330:
2326:
2322:
2316:
2312:
2306:
2302:
2298:
2289:
2281:
2276:
2270:
2267:
2255:
2245:
2228:
2225:
2221:
2194:
2191:
2168:
2043:
2039:
2033:
2026:
2022:
2018:
2015:
1877:
1820:
1819:Likewise, a
1818:
1802:
1783:
1773:
1769:
1750:
1641:
1635:
1631:
1624:
1620:
1616:
1610:
1607:
1600:
1596:
1591:
1587:
1582:
1578:
1563:
1439:
1388:
1384:
1377:
1373:
1367:
1363:
1358:
1354:
1345:
1338:
1333:
1325:
1307:
1284:
1142:
1080:group action
1079:
1075:
1054:
1036:permutations
1030:
1024:acts on any
1019:
1015:
1009:
1003:
994:
980:
976:
962:is called a
960:vector space
957:
943:acts on the
922:
916:
906:
896:
887:
883:group action
882:
881:Formally, a
880:
875:
867:group action
866:
840:
825:cyclic group
647:
635:
623:
611:
599:
587:
575:
563:
334:
291:
278:
267:
256:
252:Cyclic group
130:
117:Free product
88:Group action
87:
51:Group theory
46:Group theory
45:
8309:23 February
8266:Maskit 1988
8203:19 December
7813:Every free
7775:are called
7754:isomorphism
7444:Given left
7356:quaternions
7056:phase space
6860:cross ratio
6375:conjugation
6158:on any set
5932:) ⋅ 3
5889:) ⋅ 3
5856:) ⋅ 2
5813:) ⋅ 2
5735:denote its
5107:belongs to
4992:; that is,
4968:. Applying
4637:" or that "
4520:-invariants
4507:is denoted
4461:element of
4335:restricting
4171:orbit space
3745:irreducible
3522:the set of
3483:defined by
3465:if the map
3436:if the map
3416:Now assume
2963:unit sphere
2945:∖ {0}
2940:on the set
2853:-transitive
2666:-transitive
2368:semiregular
1811:to itself.
1343:the action
1040:cardinality
920:to itself.
885:of a group
843:mathematics
537:Topological
376:alternating
8750:Categories
8660:0873.57001
8643:2016-02-08
8597:0627.30039
8581:1180.57001
8413:References
8323:M. Artin,
8198:Proof Wiki
8114:Gain graph
7994:invertible
7777:isomorphic
7663:such that
7331:and every
7246:, but not
6960:for every
6933:such that
6878:isometries
6750:Lie groups
6534:involution
5934:| = 2
5858:| = 3
5690:which are
5453:induces a
4866:, and let
4769:This is a
4551:cohomology
4477:such that
4459:-invariant
4447:transitive
4349:is called
4061:transitive
3761:direct sum
3756:semisimple
3581:such that
3532:such that
3434:continuous
3375:such that
3359:is called
3313:for every
3298:such that
3216:such that
3017:singletons
2991:is called
2816:= 1, ...,
2787:such that
2649:-torsor.
2536:transitive
2532:is called
2055:Identity:
1886:Identity:
1308:The group
1155:Identity:
1046:Definition
951:, and the
941:polyhedron
937:symmetries
644:Symplectic
584:Orthogonal
541:Lie groups
448:Free group
173:continuous
112:Direct sum
8737:MathWorld
8720:EMS Press
8145:Citations
7926:metalogic
7633:-sets, a
7284:power set
6911:morphisms
6611:given by
6556:{1, ...,
6500:on a set
6038:∈
6031:∑
5762:⋅ 1
5534:⋅
5455:bijection
5431:over any
5335:∈
5328:⟺
5314:∈
5303:−
5294:⟺
5280:⋅
5268:−
5259:⟺
5251:⋅
5237:⋅
5229:⟺
5158:given by
5076:has type
5044:conjugate
4741:⋅
4728:∈
4695:that fix
4553:group of
4417:on which
3971:partition
3928:∈
3915:⋅
3898:⋅
3858:to which
3667:Lie group
3614:∖ {
3362:cocompact
3322:∖ {
3244:given by
3054:if every
3052:wandering
2997:partition
2993:primitive
2388:for some
2353:injective
2292:effective
2137:⋅
2123:⋅
2112:⋅
2069:⋅
1979:α
1949:α
1943:α
1896:α
1857:→
1851:×
1845::
1842:α
1779:bijection
1732:⋅
1706:⋅
1695:⋅
1660:⋅
1531:α
1506:α
1502:∘
1493:α
1465:α
1452:α
1404:α
1248:α
1224:α
1212:α
1165:α
1122:→
1116:×
1110::
1107:α
991:dimension
968:subgroups
933:triangles
872:structure
859:rotations
835:rotations
708:Conformal
596:Euclidean
203:nilpotent
8771:Symmetry
8490:(2002),
8422:(2000).
8256:, p. 72.
8232:, p. 73.
8108:See also
8032:groupoid
7918:category
7841:acts on
7829:, where
7702:and all
7690:for all
7654: :
7635:morphism
7627:are two
7587:groupoid
7542:). This
7529:(where "
7485: :
7309: :
6924: :
6895:category
6779:acts on
6467:for all
6320:for all
6247:for all
6190:and all
6178:for all
6139:Examples
5719:Example:
5640:Example:
5518:, gives
5149: :
4983:)⋅
4951:⋅(
4827:for all
4771:subgroup
4491:for all
4384:and all
4372:for all
4260: :
4202:quotient
4163:quotient
3595:for all
3384:⋅
3249:⋅(
2867:Examples
2849:sharply
2688:-tuples
2569:⋅
2564:so that
2434:embedded
2310:for all
2284:faithful
2198:acts on
2181:and all
2169:for all
1362: :
1297:and all
1285:for all
1095:function
945:vertices
927:acts on
703:Poincaré
548:Solenoid
420:Integers
410:Lattices
385:sporadic
380:Lie type
208:solvable
198:dihedral
183:additive
168:infinite
78:Subgroup
8722:, 2001
8696:0889050
8512:1867354
8442:1777008
8325:Algebra
8077:Gallery
8067:schemes
8059:actions
8016:to the
8006:to the
7998:functor
7942:monoids
7928:, this
7585:action
7364:versors
7362:1 (the
7282:on the
7122:−
6867:PGL(2,
6852:PGL(2,
6812:regular
6653:, ...,
6630:, ...,
6148:trivial
5496:⋅
5477:⋅
5442:⋅
5188:⋅
5167:⋅
5034:⋅
4963:⋅
4955:⋅
4881:⋅
4616:⋅
4571:functor
4569:of the
4541:-module
4528:. When
4482:⋅
4363:⋅
4312:⋅
4298:⋅
4256:⋅
4245:⋅
4240:, then
4094:⋅
4080:⋅
4051:⋅
4043:⋅
4013:⋅
3969:form a
3881:⋅
3775:In the
3739:over a
3702:⋅
3641:⋅
3586:⋅
3537:⋅
3501:⋅
3303:⋅
3274:of the
3221:⋅
3196:subset
3194:compact
3142:⋅
3088:⋅
2792:⋅
2713:, ...,
2697:, ...,
2597:regular
2379:⋅
2258:induces
2042:⋅
1772:⋅
1634:⋅
998:over a
970:of the
849:form a
698:Lorentz
620:Unitary
519:Lattice
459:PSL(2,
193:abelian
104:(Semi-)
8694:
8684:
8658:
8619:
8595:
8579:
8569:
8550:
8531:
8510:
8500:
8476:
8457:
8440:
8430:
8362:
8300:
8057:, and
7906:-set.)
7763:-sets
7568:-sets.
7515:α
7511:α
7483:α
7450:-sets
7377:= cos
7080:is in
6862:; the
6748:) are
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3295:U
3289:X
3285:x
3265:)
3263:y
3259:x
3255:y
3251:x
3247:n
3240:R
3234:Z
3227:K
3223:K
3219:g
3213:G
3209:g
3203:X
3199:K
3177:Ω
3172:G
3166:X
3159:G
3148:U
3144:U
3140:g
3134:G
3130:g
3124:x
3120:U
3114:G
3108:X
3104:x
3094:U
3090:U
3086:g
3080:G
3076:g
3070:U
3061:X
3057:x
3040:G
3030:X
3008:G
3002:X
2988:X
2982:G
2954:v
2943:V
2937:V
2923:n
2921:(
2915:n
2913:(
2904:n
2898:X
2892:X
2886:n
2880:n
2874:X
2860:X
2851:n
2830:n
2824:X
2818:n
2814:i
2807:i
2803:y
2798:i
2794:x
2790:g
2784:G
2780:g
2774:j
2770:i
2763:j
2759:y
2754:i
2750:y
2743:j
2739:x
2734:i
2730:x
2724:X
2719:n
2715:y
2711:1
2708:y
2703:n
2699:x
2695:1
2692:x
2690:(
2685:n
2679:n
2673:X
2664:n
2655:n
2646:G
2640:G
2630:G
2624:X
2618:g
2612:X
2608:y
2604:x
2575:y
2571:x
2567:g
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2529:X
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2505:Z
2499:Z
2495:Z
2491:5
2489:A
2484:5
2482:S
2475:2
2470:Z
2466:Z
2457:n
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2451:2
2447:)
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2441:Z
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2422:X
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2342:G
2331:G
2327:e
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2317:X
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2307:x
2303:x
2301:⋅
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2234:G
2229:g
2226:h
2220:(
2216:h
2212:g
2208:g
2204:h
2200:x
2187:X
2183:x
2179:G
2175:h
2171:g
2150:)
2147:h
2144:g
2141:(
2133:x
2130:=
2127:h
2119:)
2116:g
2108:x
2105:(
2079:x
2076:=
2073:e
2065:x
2044:g
2040:x
2029:)
2027:g
2023:x
2021:(
1997:)
1994:h
1991:g
1988:,
1985:x
1982:(
1976:=
1973:)
1970:h
1967:,
1964:)
1961:g
1958:,
1955:x
1952:(
1946:(
1917:x
1914:=
1911:)
1908:e
1905:,
1902:x
1899:(
1863:,
1860:X
1854:G
1848:X
1829:X
1825:G
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1805:)
1803:X
1797:G
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1789:G
1784:g
1774:x
1770:g
1765:x
1761:X
1757:G
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1736:x
1728:)
1725:h
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1719:(
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1667:=
1664:x
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1627:)
1625:x
1621:g
1619:(
1592:h
1583:g
1566:∘
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1515:)
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1486:=
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1461:(
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1419:)
1416:x
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1385:g
1378:g
1368:X
1364:X
1359:g
1332:-
1330:G
1322:G
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1314:X
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1221:,
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1215:(
1186:x
1183:=
1180:)
1177:x
1174:,
1171:e
1168:(
1128:,
1125:X
1119:X
1113:G
1091:X
1087:G
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1020:n
1016:S
1004:K
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981:K
977:n
917:S
907:G
897:S
888:G
830:3
828:C
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794:v
690:8
688:E
682:7
680:E
674:6
672:E
666:4
664:F
658:2
656:G
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648:n
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624:n
614:)
612:n
602:)
600:n
590:)
588:n
578:)
576:n
566:)
564:n
506:)
493:Z
481:)
468:Z
444:)
431:Z
422:(
335:p
300:Q
292:n
289:D
279:n
276:A
268:n
265:S
257:n
254:Z
31:.
20:)
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