8119:
8107:
820:
3772:
59:
5727:
5360:
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
6906:. 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
6098:
8118:
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
5195:
8106:
2007:
1276:
2160:
1746:
3947:
4767:
1873:
1138:
5622:
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:
5977:
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
8001:
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
7625:
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.
5934:
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
5854:
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
8812:
6001:
364:
8732:
8667:
8617:
8598:
8548:
8524:
8478:
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
8807:
5766:, and this action is transitive as can be seen by composing rotations about the center of the cube. Thus, by the orbit-stabilizer theorem,
6124:
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.
5514:
314:
6591:
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:
8368:
799:
309:
8579:
8505:
8410:
8323:
4704:
7973:
Instead of actions on sets, we can define actions of groups and monoids on objects of an arbitrary category: start with an object
7452:
5520:
725:
8313:
8155:
7966:
on sets, by using the same two axioms as above. This does not define bijective maps and equivalence relations however. See
1938:
1207:
8766:
8385:
6383:
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
8199:
8181:
5739:
8761:
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:
8756:
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:
8677:
8039:
4813:
4334:
3390:
2839:
1350:
846:
6697:, or group, or ring ...) acts on the vector space (or set of vertices of the graph, or group, or ring ...).
5717:
This result is especially useful since it can be employed for counting arguments (typically in situations where
5355:{\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}.}
6555:
607:
341:
218:
106:
28:
8145:
8060:
6872:. This is a quotient of the action of the general linear group on projective space. Particularly notable is
6844:
6808:
3676:
3411:
3016:
966:
of the group. In the case of a finite-dimensional vector space, it allows one to identify many groups with
8049:
6916:
3361:
2257:
1094:
871:
858:
834:
819:
757:
547:
8802:
6687:
3351:
2428:
For example, the action of any group on itself by left multiplication is free. This observation implies
631:
1398:
8140:
8124:
Orbit of a fundamental spherical triangle (marked in red) under action of the full icosahedral group.
8088:
8002:
7381:
6832:, and the subgroup V of the affine group (that is, a vector space) has transitive and free (that is,
5631:
5116:
3980:
3736:
3724:
2948:
2931:
1569:
971:
963:
911:
854:
571:
559:
177:
111:
8638:, Grundlehren der Mathematischen Wissenschaften, vol. 287, Springer-Verlag, pp. XIII+326,
8112:
Orbit of a fundamental spherical triangle (marked in red) under action of the full octahedral group.
1891:
1160:
8719:, de Gruyter Studies in Mathematics, vol. 8, Berlin: Walter de Gruyter & Co., p. 29,
8080:
7085:
6780:. The group operations are given by multiplying the matrices from the groups with the vectors from
6288:
5995:
5411:
3271:
2429:
1573:
1060:
999:
986:
850:
146:
41:
2060:
1651:
487:
462:
425:
8466:
7943:
7581:
6396:
6291:– that every group is isomorphic to a subgroup of the symmetric group of permutations of the set
5753:
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.
8817:
8775:
8728:
8663:
8613:
8594:
8575:
8544:
8520:
8501:
8474:
8406:
8364:
8319:
8076:
7089:
6225:
6154:
4023:
3786:
3771:
3423:
3275:
3034:
2908:
1025:
944:
892:
702:
536:
379:
273:
8215:
1554:{\displaystyle \alpha _{g}(\alpha _{h}(x))=(\alpha _{g}\circ \alpha _{h})(x)=\alpha _{gh}(x)}
8720:
8702:
8655:
8639:
8623:
8064:
8029:
7967:
7093:
6859:
6756:
6729:
6598:
6367:
4550:
4209:
3740:
3712:
3451:
2958:
1064:
687:
679:
671:
663:
655:
643:
583:
523:
513:
355:
297:
172:
141:
8742:
8701:, Princeton Mathematical Series, vol. 35, Princeton University Press, pp. x+311,
8558:
8488:
8738:
8706:
8643:
8627:
8554:
8484:
8072:
8015:
7939:
7744:
7370:
7002:
6903:
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:
6885:
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
6858:
and its subgroups, particularly its Lie subgroups, which are Lie groups that act on the
6287:. 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
6880:, the symmetries of the projective line, which is sharply 3-transitive, preserving the
6588:
6150:
4213:
3993:
3554:
3279:
2241:
862:
778:
714:
404:
384:
321:
286:
207:
197:
182:
167:
121:
98:
6594:
The symmetry group of any geometrical object acts on the set of points of that object.
8796:
8538:
8534:
8150:
7986:
7366:
6140:
5743:
5446:
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:
5634:. In particular that implies that the orbit length is a divisor of the group order.
8092:
8084:
7081:
6998:
6837:
6829:
6821:
6713:
6694:
6529:
2461:
2336:
959:
824:
552:
251:
240:
187:
162:
157:
116:
50:
8056:
is a functor from the groupoid to the category of sets or to some other category.
5726:
4449:
if and only if all elements are equivalent, meaning that there is only one orbit.
8778:
8714:
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
7775:
7077:
7066:
6881:
2962:
1039:
1035:
842:
8659:
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
8724:
8135:
7377:
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:
6836:) 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,
8783:
7947:
7305:
6771:
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
17:
7570:-set has the property that its fixed points correspond to equivariant maps
7166:
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:
8068:
8053:
7608:
7600:
6932:
6899:
6807:
by natural matrix action. The orbits of its action are classified by the
6425:. 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:
5738:
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
8019:
7922:
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:
8683:
8363:(1st ed.). The Mathematical Association of America. p. 200.
8063:
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
7963:
7385:
4231:
1144:
7761:
The composition of two morphisms is again a morphism. If a morphism
7365:. This is useful, for instance, in studying the action of the large
4208:. The coinvariant terminology and notation are used particularly in
8405:. Cambridge, UK New York: Cambridge University Press. p. 170.
7951:
5725:
5396:
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
7767:
is bijective, then its inverse is also a morphism. In this case
8052:
between the group action functors. In analogy, an action of a
6902:
of the plane act on the set of 2D images and patterns, such as
5862:, which permutes 2, 4, 5 and 3, 6, 8, and fixes 1 and 7. Thus,
8612:, Modern Birkhäuser Classics, Birkhäuser, pp. xxvii+467,
8315:
Lie Groups: An
Approach through Invariants and Representations
8014:
as a category with a single object in which every morphism is
5971:
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.
6093:{\displaystyle |X/G|={\frac {1}{|G|}}\sum _{g\in G}|X^{g}|,}
5994:
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
5513:
is finite then the orbit-stabilizer theorem, together with
5192:. 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.
8018:. A (left) group action is then nothing but a (covariant)
7459:
leaves all points where they were, as does the quaternion
7369:
on a 24-set and in studying symmetry in certain models of
7020:
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
5131:
Orbits and stabilizers are closely related. For a fixed
2047:
when the action being considered is clear from context)
7455:. 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.
8682:, 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.
6004:
5523:
5198:
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:
8318:. Springer Science & Business Media. p. 5.
7932:
With this notion of morphism, the collection of all
6538:, given by the action of 1. Similarly, an action of
5471:
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}
6825:
6092:
5616:
5354:
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:
7873:by left multiplication on the first coordinate. (
7605:The notion of group action can be encoded by the
7050:, 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
8699:Three-dimensional geometry and topology. Vol. 1.
8095:acting on objects of their respective category.
7433:is a counterclockwise rotation through an angle
2621:in the definition of transitivity is unique. If
8032:, and a group representation is a functor from
7979:of some category, and then define an action on
7898: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\}.}
6693:The automorphism group of a vector space (or
5983:| = 8 ⋅ 3 ⋅ 2 ⋅ 1 = 48
5801:. 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:
8679:The geometry and topology of three-manifolds
7985:as a monoid homomorphism into the monoid of
6528: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
7904:on the set of left cosets of some subgroup
6224:; that is, every group element induces the
4565:, and the higher cohomology groups are the
6382:to a subgroup of the permutation group of
6133:, the set of formal differences of finite
5877:. Applying the theorem a third time gives
5624:in other words the length of the orbit of
5329:
5325:
5295:
5291:
5260:
5256:
5230:
5226:
3278:of a locally simply connected space on an
2240:can be considered as a left action of its
807:
793:
245:
71:
36:
8299:
7801:; for all practical purposes, isomorphic
7130:is defined to be the state of the system
6082:
6076:
6067:
6055:
6043:
6035:
6029:
6021:
6013:
6005:
6003:
5707:-invariant elements are congruent modulo
5630:times the order of its stabilizer is the
5606:
5600:
5591:
5586:
5581:
5573:
5561:
5556:
5552:
5538:
5524:
5522:
5343:
5319:
5300:
5277:
5265:
5248:
5234:
5197:
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:
8610:Hyperbolic manifolds and discrete groups
8263:
8251:
8239:
5695:-invariant elements. More specifically,
5617:{\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
8593:. Textbooks in mathematics. CRC Press.
8568:An Introduction to the Theory of Groups
8449:
8437:
8425:
8275:
8172:
8102:
7158: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:
8287:
8216:"Definition:Right Group Action Axioms"
7820:action is isomorphic to the action of
6312:, left multiplication is an action of
6243:, left multiplication is an action of
5665: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
8515:Eie, Minking; Chang, Shou-Te (2010).
8496:Dummit, David; Richard Foote (2003).
7879:can be taken to be the set of orbits
7439:about an axis given by a unit vector
7388:), as a multiplicative group, act on
7298:, we can define an induced action of
7120: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:
7558:" indicates right multiplication by
6716:(including the special linear group
6712:and its subgroups, particularly its
6112:. 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
7629:Morphisms and isomorphisms between
5730: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:
8757:"Action of a group on a manifold"
8574:(4th ed.). Springer-Verlag.
6576:and its subgroups act on the set
6376: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
8591:Introduction to abstract algebra
8570:. Graduate Texts in Mathematics
8201:Introduction to abstract algebra
8117:
8105:
7962:We can also consider actions of
7453:quaternions and spatial rotation
6811:of coordinates of the vector in
6554:is equivalent to the data of an
6149:, 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:
8813:Representation theory of groups
8091:. All of these are examples of
7946:(in fact, assuming a classical
6925:-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
8543:, Cambridge University Press,
8473:. Cambridge University Press.
8196:This is done, for example, by
6941:-set homomorphisms: functions
6463:, conjugation is an action of
6139:-sets forms a ring called the
6106:is the set of points fixed by
6083:
6068:
6044:
6036:
6022:
6006:
5607:
5592:
5582:
5574:
5567:
5546:
5539:
5525:
5500:. This result is known as the
5326:
5292:
5257:
5227:
5223:
5217:
5208:
5202:
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:
8697:Thurston, William P. (1997),
8589:Smith, Jonathan D.H. (2008).
8349:, Proposition 6.8.4 on p. 179
7832:given by left multiplication.
7807:-sets are indistinguishable.
6909:The sets acted on by a group
6774:that act on the vector space
5675:elements, there are at least
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
8517:A Course on Abstract Algebra
8387:A Course on Abstract Algebra
8183:A Course on Abstract Algebra
7958:Variants and generalizations
7601:Groupoid § Group action
7114:is in the phase space, then
5762:acts on the set of vertices
5399: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} }
8808:Group actions (mathematics)
8762:Encyclopedia of Mathematics
7810:Some example isomorphisms:
7595:Group actions and groupoids
7580:; more generally, it is an
7038:correspond to subfields of
6743:, 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
8834:
8676:Thurston, William (1980),
8660:10.1142/9789811286018_0005
8608:Kapovich, Michael (2009),
7598:
7451:is the same rotation; see
7394:: for any such quaternion
7065:The additive group of the
6895:is of particular interest.
5647:be a group of prime order
5171:. 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:
8725:10.1515/9783110858372.312
8713:tom Dieck, Tammo (1987),
8312:Procesi, Claudio (2007).
8156:Young–Deruyts development
8040:category of vector spaces
6789:The general linear group
6700:The general linear group
6686:is a group action called
6584:by permuting its elements
5182: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
8652:Starting Category Theory
8634:Maskit, Bernard (1988),
8384:Eie & Chang (2010).
8180:Eie & Chang (2010).
7841:action is isomorphic to
7286:Given a group action of
6153:, and multiplication to
5502: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)
8650:Perrone, Paolo (2024),
8566:Rotman, Joseph (1995).
8500:(3rd ed.). Wiley.
8359:Carter, Nathan (2009).
8242:, Definition 3.5.1(iv).
8146:Measurable group action
7954:will even be Boolean).
7182:equal to, for example,
6845:projective linear group
6809:greatest common divisor
6366: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
8050:natural transformation
7742:-sets are also called
6094:
5731:
5618:
5356:
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:
8716:Transformation groups
8403:Geometry and topology
8071:, regular actions of
8042:. A morphism between
8003:group representations
7942:; this category is a
7088:(and in more general
6688:scalar multiplication
6318:on the set of cosets
6095:
5989:
5953:. One also sees that
5746:as pictured, and let
5729:
5619:
5357:
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:
8654:, World Scientific,
8519:. World Scientific.
8401:Reid, Miles (2005).
8141:Group with operators
8008:We can view a group
6828:on the points of an
6612:with group of units
6567:The symmetric group
6226:identity permutation
6174:action of any group
6002:
5723:is finite as well).
5521:
5196:
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:
8471:Finite Group Theory
8467:Aschbacher, Michael
8361:Visual Group Theory
8077:algebraic varieties
7584:in the category of
7496:whose elements are
7086:classical mechanics
6360:. In particular if
5143:, 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
8776:Weisstein, Eric W.
8540:Algebraic Topology
8061:continuous actions
7944:Grothendieck topos
7582:exponential object
7502:-equivariant maps
7484:, there is a left
7014:acts on the field
6904:wallpaper patterns
6090:
6066:
5732:
5701:and the number of
5632:order of the group
5614:
5515:Lagrange's theorem
5352:
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
8734:978-3-11-009745-0
8669:978-981-12-8600-1
8619:978-0-8176-4912-8
8600:978-1-4200-6371-4
8550:978-0-521-79540-1
8526:978-981-4271-88-2
8480:978-0-521-78675-1
8290:, II.A.1, II.A.2.
8067:of Lie groups on
8005:in this fashion.
7892:Every transitive
7526:-action given by
7371:finite geometries
7341:for every subset
7136:seconds later if
7090:dynamical systems
6469:on conjugates of
6155:Cartesian product
6051:
6049:
5928:. Any element of
5848:. Any element of
5790:| = 8 |
5685: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
8825:
8789:
8788:
8770:
8745:
8709:
8693:
8692:
8691:
8672:
8646:
8630:
8604:
8585:
8561:
8530:
8511:
8498:Abstract Algebra
8492:
8453:
8452:, pp. 69–71
8447:
8441:
8440:, pp. 36–39
8435:
8429:
8423:
8417:
8416:
8398:
8392:
8391:
8381:
8375:
8374:
8356:
8350:
8343:
8337:
8336:
8334:
8332:
8309:
8303:
8297:
8291:
8285:
8279:
8273:
8267:
8261:
8255:
8249:
8243:
8237:
8231:
8230:
8228:
8226:
8212:
8206:
8205:
8194:
8188:
8187:
8177:
8121:
8109:
8073:algebraic groups
8069:smooth manifolds
8048:-sets is then a
8047:
8037:
8030:category of sets
8027:
8013:
8000:
7994:
7984:
7978:
7968:semigroup action
7937:
7927:
7921:
7915:
7909:
7903:
7897:
7888:
7878:
7872:
7862:
7857:is some set and
7856:
7850:
7840:
7831:
7825:
7819:
7806:
7796:
7790:
7784:
7772:
7766:
7753:
7745:equivariant maps
7741:
7735:
7729:
7723:
7717:
7711:
7684:
7670:
7664:
7654:
7648:
7642:
7624:
7589:
7579:
7569:
7563:
7557:
7550:
7525:
7520:, and with left
7519:
7501:
7495:
7489:
7483:
7477:
7471:
7462:
7458:
7450:
7444:
7438:
7432:
7412:
7393:
7364:
7358:
7352:
7346:
7340:
7313:
7303:
7297:
7291:
7282:
7267:
7253:
7235:
7221:
7210:
7196:
7181:
7165:
7154:
7148:
7141:
7135:
7129:
7119:
7113:
7107:
7101:
7094:time translation
7075:
7061:
7055:
7049:
7043:
7037:
7025:
7019:
7013:
6993:
6987:
6981:
6954:
6940:
6930:
6924:
6914:
6894:
6879:
6871:
6860:projective space
6857:
6816:
6806:
6800:
6785:
6779:
6769:
6757:symplectic group
6754:
6742:
6730:orthogonal group
6727:
6711:
6685:
6632:
6618:
6611:
6605:
6599:coordinate space
6583:
6575:
6563:
6553:
6547:
6537:
6527:
6521:
6512:
6506:
6500:
6494:
6488:
6474:
6468:
6462:
6456:
6447:
6438:; it satisfies (
6437:
6424:
6410:
6404:
6399:is an action of
6394:
6381:
6375:
6368:normal subgroups
6365:
6359:
6353:
6347:
6341:
6327:
6317:
6311:
6305:
6296:
6289:Cayley's theorem
6286:
6280:
6274:
6268:
6254:
6248:
6242:
6233:
6223:
6217:
6211:
6205:
6199:
6185:
6179:
6172:
6171:
6148:
6138:
6132:
6123:
6117:
6111:
6105:
6099:
6097:
6096:
6091:
6086:
6081:
6080:
6071:
6065:
6050:
6048:
6047:
6039:
6030:
6025:
6017:
6009:
5996:Burnside's lemma
5990:Burnside's lemma
5984:
5982:
5976:
5970:
5952:
5950:
5933:
5927:
5925:
5907:
5892:
5876:
5874:
5861:
5853:
5847:
5845:
5831:
5820:
5810:, we can obtain
5809:
5800:
5798:
5789:
5780:
5773:
5765:
5761:
5751:
5722:
5712:
5706:
5700:
5694:
5688:
5684:
5674:
5668:
5664:
5658:
5653:acting on a set
5652:
5646:
5629:
5623:
5621:
5620:
5615:
5610:
5605:
5604:
5595:
5590:
5585:
5577:
5566:
5565:
5542:
5528:
5512:
5499:
5480:
5470:
5456:between the set
5451:
5445:
5435:
5429:
5423:
5409:
5395:lie in the same
5394:
5388:
5380:
5362:In other words,
5361:
5359:
5358:
5353:
5348:
5347:
5324:
5323:
5308:
5307:
5281:
5273:
5272:
5252:
5238:
5191:
5181:
5170:
5156:
5142:
5136:
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:
8833:
8832:
8828:
8827:
8826:
8824:
8823:
8822:
8793:
8792:
8774:
8773:
8755:
8752:
8735:
8712:
8696:
8689:
8687:
8675:
8670:
8649:
8636:Kleinian groups
8633:
8620:
8607:
8601:
8588:
8582:
8565:
8551:
8533:
8527:
8514:
8508:
8495:
8481:
8465:
8462:
8457:
8456:
8448:
8444:
8436:
8432:
8424:
8420:
8413:
8400:
8399:
8395:
8383:
8382:
8378:
8371:
8358:
8357:
8353:
8344:
8340:
8330:
8328:
8326:
8311:
8310:
8306:
8298:
8294:
8286:
8282:
8274:
8270:
8262:
8258:
8250:
8246:
8238:
8234:
8224:
8222:
8214:
8213:
8209:
8197:
8195:
8191:
8179:
8178:
8174:
8169:
8164:
8132:
8125:
8122:
8113:
8110:
8101:
8059:In addition to
8043:
8033:
8023:
8009:
7996:
7990:
7980:
7974:
7960:
7933:
7923:
7917:
7911:
7905:
7899:
7893:
7880:
7874:
7864:
7858:
7852:
7842:
7836:
7827:
7821:
7815:
7802:
7792:
7786:
7780:
7768:
7762:
7749:
7737:
7736:. Morphisms of
7731:
7725:
7719:
7713:
7686:
7672:
7666:
7660:
7650:
7644:
7638:
7635:
7612:
7603:
7597:
7585:
7571:
7565:
7559:
7552:
7544:
7527:
7521:
7503:
7497:
7491:
7485:
7479:
7473:
7467:
7460:
7456:
7446:
7440:
7434:
7414:
7395:
7389:
7360:
7354:
7348:
7342:
7315:
7309:
7299:
7293:
7287:
7269:
7255:
7237:
7223:
7212:
7198:
7183:
7167:
7159:
7150:
7149:seconds ago if
7143:
7142:is positive or
7137:
7131:
7121:
7115:
7109:
7103:
7097:
7069:
7057:
7051:
7045:
7039:
7027:
7026:. Subgroups of
7021:
7015:
7005:
7003:field extension
6989:
6983:
6956:
6942:
6936:
6926:
6920:
6910:
6888:
6873:
6862:
6847:
6812:
6802:
6790:
6781:
6775:
6759:
6744:
6732:
6717:
6701:
6683:
6674:
6667:
6660:
6651:
6644:
6634:
6620:
6613:
6607:
6601:
6577:
6574:
6568:
6559:
6549:
6539:
6533:
6523:
6517:
6508:
6502:
6496:
6490:
6476:
6470:
6464:
6458:
6452:
6451:In every group
6439:
6426:
6412:
6406:
6400:
6390:
6389:In every group
6377:
6371:
6361:
6355:
6349:
6343:
6329:
6319:
6313:
6307:
6301:
6300:In every group
6292:
6282:
6276:
6270:
6256:
6250:
6244:
6238:
6237:In every group
6229:
6219:
6213:
6207:
6201:
6187:
6181:
6175:
6169:
6168:
6163:
6144:
6134:
6128:
6127:Fixing a group
6119:
6113:
6107:
6101:
6072:
6034:
6000:
5999:
5992:
5980:
5978:
5972:
5969:
5965:
5961:
5954:
5948:
5944:
5937:
5935:
5929:
5924:
5920:
5916:
5909:
5905:
5901:
5894:
5893:| = |
5891:
5887:
5880:
5878:
5872:
5865:
5863:
5855:
5849:
5844:
5840:
5833:
5829:
5822:
5821:| = |
5819:
5813:
5811:
5808:
5802:
5797:
5791:
5788:
5782:
5775:
5774:| = |
5769:
5767:
5763:
5757:
5747:
5742:. Consider the
5718:
5708:
5702:
5696:
5690:
5686:
5676:
5670:
5666:
5660:
5654:
5648:
5642:
5625:
5596:
5557:
5519:
5518:
5508:
5490:
5482:
5472:
5469:
5457:
5447:
5437:
5431:
5425:
5414:
5408:
5400:
5390:
5384:
5363:
5339:
5315:
5296:
5261:
5194:
5193:
5183:
5172:
5158:
5144:
5138:
5132:
5129:
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:
8831:
8829:
8821:
8820:
8815:
8810:
8805:
8795:
8794:
8791:
8790:
8779:"Group Action"
8771:
8751:
8750:External links
8748:
8747:
8746:
8733:
8710:
8694:
8673:
8668:
8647:
8631:
8618:
8605:
8599:
8586:
8580:
8563:
8549:
8535:Hatcher, Allen
8531:
8525:
8512:
8506:
8493:
8479:
8461:
8458:
8455:
8454:
8450:Perrone (2024)
8442:
8438:Perrone (2024)
8430:
8428:, pp. 7–9
8426:Perrone (2024)
8418:
8411:
8393:
8390:. p. 145.
8376:
8370:978-0883857571
8369:
8351:
8338:
8324:
8304:
8300:tom Dieck 1987
8292:
8280:
8268:
8266:, p. 176.
8256:
8244:
8232:
8207:
8204:. p. 253.
8198:Smith (2008).
8189:
8186:. p. 144.
8171:
8170:
8168:
8165:
8163:
8160:
8159:
8158:
8153:
8148:
8143:
8138:
8131:
8128:
8127:
8126:
8123:
8116:
8114:
8111:
8104:
8100:
8097:
8065:smooth actions
7959:
7956:
7938:-sets forms a
7930:
7929:
7890:
7833:
7814:Every regular
7779:, and the two
7671:is a function
7634:
7627:
7599:Main article:
7596:
7593:
7592:
7591:
7540:
7464:
7413:, the mapping
7374:
7284:
7156:
7063:
6995:
6931:-sets and the
6907:
6896:
6841:
6818:
6787:
6698:
6691:
6679:
6672:
6665:
6656:
6649:
6642:
6619:, the mapping
6595:
6592:
6589:symmetry group
6585:
6570:
6565:
6514:
6507:conjugates of
6457:with subgroup
6449:
6387:
6306:with subgroup
6298:
6235:
6186:is defined by
6162:
6159:
6151:disjoint union
6089:
6085:
6079:
6075:
6070:
6064:
6061:
6058:
6054:
6046:
6042:
6038:
6033:
6028:
6024:
6020:
6016:
6012:
6008:
5991:
5988:
5987:
5986:
5967:
5963:
5959:
5946:
5942:
5922:
5918:
5914:
5903:
5899:
5889:
5885:
5870:
5842:
5838:
5827:
5817:
5806:
5795:
5786:
5764:{1, 2, ..., 8}
5715:
5714:
5613:
5609:
5603:
5599:
5594:
5589:
5584:
5580:
5576:
5572:
5569:
5564:
5560:
5555:
5551:
5548:
5545:
5541:
5537:
5534:
5531:
5527:
5486:
5481:, which sends
5465:
5455:
5404:
5382:if and only if
5351:
5346:
5342:
5338:
5335:
5332:
5328:
5322:
5318:
5314:
5311:
5306:
5303:
5299:
5294:
5290:
5287:
5284:
5280:
5276:
5271:
5268:
5264:
5259:
5255:
5251:
5247:
5244:
5241:
5237:
5233:
5229:
5225:
5222:
5219:
5216:
5213:
5210:
5207:
5204:
5201:
5128:
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:
8830:
8819:
8816:
8814:
8811:
8809:
8806:
8804:
8801:
8800:
8798:
8786:
8785:
8780:
8777:
8772:
8768:
8764:
8763:
8758:
8754:
8753:
8749:
8744:
8740:
8736:
8730:
8726:
8722:
8718:
8717:
8711:
8708:
8704:
8700:
8695:
8686:on 2020-07-27
8685:
8681:
8680:
8674:
8671:
8665:
8661:
8657:
8653:
8648:
8645:
8641:
8637:
8632:
8629:
8625:
8621:
8615:
8611:
8606:
8602:
8596:
8592:
8587:
8583:
8581:0-387-94285-8
8577:
8573:
8569:
8564:
8560:
8556:
8552:
8546:
8542:
8541:
8536:
8532:
8528:
8522:
8518:
8513:
8509:
8507:0-471-43334-9
8503:
8499:
8494:
8490:
8486:
8482:
8476:
8472:
8468:
8464:
8463:
8459:
8451:
8446:
8443:
8439:
8434:
8431:
8427:
8422:
8419:
8414:
8412:9780521613255
8408:
8404:
8397:
8394:
8389:
8388:
8380:
8377:
8372:
8366:
8362:
8355:
8352:
8348:
8342:
8339:
8327:
8325:9780387289298
8321:
8317:
8316:
8308:
8305:
8301:
8296:
8293:
8289:
8284:
8281:
8277:
8272:
8269:
8265:
8264:Thurston 1980
8260:
8257:
8253:
8252:Kapovich 2009
8248:
8245:
8241:
8240:Thurston 1997
8236:
8233:
8221:
8217:
8211:
8208:
8203:
8202:
8193:
8190:
8185:
8184:
8176:
8173:
8166:
8161:
8157:
8154:
8152:
8151:Monoid action
8149:
8147:
8144:
8142:
8139:
8137:
8134:
8133:
8129:
8120:
8115:
8108:
8103:
8098:
8096:
8094:
8093:group objects
8090:
8086:
8085:group schemes
8082:
8078:
8074:
8070:
8066:
8062:
8057:
8055:
8051:
8046:
8041:
8036:
8031:
8026:
8021:
8017:
8012:
8006:
8004:
7999:
7993:
7988:
7987:endomorphisms
7983:
7977:
7971:
7969:
7965:
7957:
7955:
7953:
7949:
7945:
7941:
7936:
7926:
7920:
7914:
7908:
7902:
7896:
7891:
7887:
7883:
7877:
7871:
7867:
7861:
7855:
7849:
7845:
7839:
7834:
7830:
7824:
7818:
7813:
7812:
7811:
7808:
7805:
7800:
7795:
7789:
7783:
7778:
7777:
7773:is called an
7771:
7765:
7759:
7757:
7752:
7747:
7746:
7740:
7734:
7728:
7722:
7716:
7709:
7705:
7701:
7697:
7693:
7689:
7683:
7679:
7675:
7669:
7663:
7658:
7653:
7647:
7641:
7632:
7628:
7626:
7623:
7619:
7615:
7611:
7610:
7602:
7594:
7588:
7583:
7578:
7574:
7568:
7562:
7556:
7548:
7543:
7538:
7534:
7530:
7524:
7518:
7514:
7510:
7506:
7500:
7494:
7488:
7482:
7476:
7470:
7465:
7454:
7449:
7443:
7437:
7431:
7428:
7425:
7421:
7417:
7410:
7406:
7402:
7398:
7392:
7387:
7383:
7379:
7375:
7372:
7368:
7367:Mathieu group
7363:
7357:
7351:
7345:
7338:
7334:
7330:
7326:
7322:
7318:
7314:, by setting
7312:
7307:
7302:
7296:
7290:
7285:
7280:
7276:
7272:
7266:
7262:
7258:
7252:
7248:
7244:
7240:
7234:
7230:
7226:
7219:
7215:
7209:
7205:
7201:
7194:
7190:
7186:
7179:
7175:
7171:
7163:
7157:
7153:
7147:
7140:
7134:
7128:
7124:
7118:
7112:
7106:
7100:
7095:
7091:
7087:
7084:" systems in
7083:
7079:
7073:
7068:
7064:
7060:
7054:
7048:
7044:that contain
7042:
7035:
7031:
7024:
7018:
7012:
7008:
7004:
7000:
6996:
6992:
6986:
6979:
6975:
6971:
6967:
6963:
6959:
6953:
6949:
6945:
6939:
6934:
6929:
6923:
6918:
6915:comprise the
6913:
6908:
6905:
6901:
6897:
6892:
6887:
6883:
6877:
6869:
6865:
6861:
6855:
6851:
6846:
6842:
6839:
6835:
6831:
6827:
6823:
6819:
6815:
6810:
6805:
6798:
6794:
6788:
6784:
6778:
6773:
6767:
6763:
6758:
6752:
6748:
6740:
6736:
6731:
6725:
6721:
6715:
6714:Lie subgroups
6709:
6705:
6699:
6696:
6692:
6689:
6682:
6678:
6671:
6664:
6659:
6655:
6648:
6641:
6637:
6631:
6627:
6623:
6616:
6610:
6606:over a field
6604:
6600:
6596:
6593:
6590:
6586:
6581:
6573:
6566:
6562:
6557:
6552:
6546:
6542:
6536:
6531:
6526:
6520:
6516:An action of
6515:
6511:
6505:
6499:
6493:
6487:
6483:
6479:
6473:
6467:
6461:
6455:
6450:
6446:
6442:
6436:
6433:
6429:
6423:
6419:
6415:
6409:
6403:
6398:
6393:
6388:
6385:
6380:
6374:
6369:
6364:
6358:
6352:
6346:
6340:
6336:
6332:
6326:
6322:
6316:
6310:
6304:
6299:
6295:
6290:
6285:
6279:
6273:
6267:
6263:
6259:
6253:
6247:
6241:
6236:
6232:
6227:
6222:
6216:
6210:
6204:
6198:
6194:
6190:
6184:
6178:
6173:
6165:
6164:
6160:
6158:
6156:
6152:
6147:
6142:
6141:Burnside ring
6137:
6131:
6125:
6122:
6116:
6110:
6104:
6087:
6077:
6073:
6062:
6059:
6056:
6052:
6040:
6031:
6026:
6018:
6014:
6010:
5997:
5975:
5958:
5941:
5932:
5913:
5908:| |
5898:
5884:
5869:
5859:
5852:
5837:
5832:| |
5826:
5816:
5805:
5794:
5785:
5781:| |
5778:
5772:
5760:
5755:
5750:
5745:
5744:cubical graph
5741:
5737:
5734:
5733:
5728:
5724:
5721:
5711:
5705:
5699:
5693:
5683:
5679:
5673:
5663:
5657:
5651:
5645:
5640:
5637:
5636:
5635:
5633:
5628:
5611:
5601:
5597:
5587:
5578:
5570:
5562:
5558:
5553:
5549:
5543:
5535:
5532:
5529:
5516:
5511:
5505:
5503:
5498:
5494:
5489:
5485:
5479:
5475:
5468:
5464:
5460:
5453:
5450:
5444:
5440:
5434:
5428:
5421:
5417:
5413:
5407:
5403:
5398:
5393:
5387:
5383:
5378:
5374:
5370:
5366:
5349:
5344:
5340:
5336:
5333:
5330:
5320:
5316:
5312:
5309:
5304:
5301:
5297:
5288:
5285:
5282:
5278:
5274:
5269:
5266:
5262:
5253:
5249:
5245:
5242:
5239:
5235:
5231:
5220:
5214:
5211:
5205:
5199:
5190:
5186:
5179:
5175:
5169:
5165:
5161:
5155:
5151:
5147:
5141:
5135:
5122:
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:
8803:Group theory
8782:
8760:
8715:
8698:
8688:, retrieved
8684:the original
8678:
8651:
8635:
8609:
8590:
8571:
8567:
8539:
8516:
8497:
8470:
8445:
8433:
8421:
8402:
8396:
8386:
8379:
8360:
8354:
8346:
8341:
8329:. Retrieved
8314:
8307:
8295:
8283:
8276:Hatcher 2002
8271:
8259:
8247:
8235:
8223:. Retrieved
8219:
8210:
8200:
8192:
8182:
8175:
8058:
8044:
8034:
8024:
8010:
8007:
7997:
7991:
7981:
7975:
7972:
7961:
7934:
7931:
7924:
7918:
7912:
7906:
7900:
7894:
7885:
7881:
7875:
7869:
7865:
7859:
7853:
7847:
7843:
7837:
7828:
7822:
7816:
7809:
7803:
7798:
7793:
7787:
7781:
7774:
7769:
7763:
7760:
7755:
7750:
7743:
7738:
7732:
7726:
7720:
7714:
7707:
7703:
7699:
7695:
7691:
7687:
7681:
7677:
7673:
7667:
7661:
7656:
7651:
7645:
7639:
7636:
7630:
7621:
7617:
7613:
7606:
7604:
7586:
7576:
7572:
7566:
7560:
7554:
7546:
7541:
7536:
7532:
7528:
7522:
7516:
7512:
7508:
7504:
7498:
7492:
7486:
7480:
7474:
7468:
7447:
7441:
7435:
7429:
7426:
7423:
7419:
7415:
7408:
7404:
7400:
7396:
7390:
7361:
7355:
7349:
7343:
7336:
7332:
7328:
7324:
7320:
7316:
7310:
7300:
7294:
7288:
7278:
7274:
7270:
7264:
7260:
7256:
7250:
7246:
7242:
7238:
7232:
7228:
7224:
7217:
7213:
7207:
7203:
7199:
7192:
7188:
7184:
7177:
7173:
7169:
7161:
7155:is negative.
7151:
7145:
7138:
7132:
7126:
7122:
7116:
7110:
7104:
7098:
7082:well-behaved
7076:acts on the
7071:
7067:real numbers
7058:
7052:
7046:
7040:
7033:
7029:
7022:
7016:
7010:
7006:
6999:Galois group
6990:
6984:
6977:
6973:
6969:
6965:
6961:
6957:
6951:
6947:
6943:
6937:
6927:
6921:
6911:
6890:
6886:Möbius group
6875:
6867:
6863:
6853:
6849:
6838:affine space
6833:
6830:affine space
6826:transitively
6822:affine group
6813:
6803:
6796:
6792:
6782:
6776:
6765:
6761:
6750:
6746:
6738:
6734:
6723:
6719:
6707:
6703:
6680:
6676:
6669:
6662:
6657:
6653:
6646:
6639:
6635:
6629:
6625:
6621:
6614:
6608:
6602:
6579:
6571:
6560:
6550:
6544:
6540:
6534:
6530:automorphism
6524:
6518:
6509:
6503:
6497:
6491:
6485:
6481:
6477:
6471:
6465:
6459:
6453:
6444:
6440:
6434:
6431:
6427:
6421:
6417:
6413:
6407:
6401:
6391:
6378:
6372:
6362:
6356:
6350:
6344:
6338:
6334:
6330:
6324:
6320:
6314:
6308:
6302:
6293:
6283:
6277:
6271:
6265:
6261:
6257:
6251:
6245:
6239:
6230:
6220:
6214:
6208:
6202:
6196:
6192:
6188:
6182:
6176:
6167:
6145:
6135:
6129:
6126:
6120:
6114:
6108:
6102:
5993:
5973:
5956:
5939:
5930:
5911:
5896:
5882:
5867:
5857:
5850:
5835:
5824:
5814:
5803:
5792:
5783:
5776:
5770:
5758:
5756:group. Then
5754:automorphism
5748:
5735:
5719:
5716:
5709:
5703:
5697:
5691:
5681:
5677:
5671:
5661:
5655:
5649:
5643:
5638:
5626:
5509:
5506:
5501:
5496:
5492:
5487:
5483:
5477:
5473:
5466:
5462:
5458:
5448:
5442:
5438:
5432:
5426:
5419:
5415:
5410:. Thus, the
5405:
5401:
5391:
5385:
5381:
5376:
5372:
5368:
5364:
5188:
5184:
5177:
5173:
5167:
5163:
5159:
5153:
5149:
5145:
5139:
5133:
5130:
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:
18:N-transitive
8331:23 February
8288:Maskit 1988
8225:19 December
7835:Every free
7797:are called
7776:isomorphism
7466:Given left
7378:quaternions
7078:phase space
6882:cross ratio
6397:conjugation
6180:on any set
5949:) ⋅ 3
5906:) ⋅ 3
5873:) ⋅ 2
5830:) ⋅ 2
5752: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
8797:Categories
8707:0873.57001
8690:2016-02-08
8644:0627.30039
8628:1180.57001
8460:References
8345:M. Artin,
8220:Proof Wiki
8136:Gain graph
8016:invertible
7799:isomorphic
7685:such that
7353:and every
7268:, but not
6982:for every
6955:such that
6900:isometries
6772:Lie groups
6556:involution
5951:| = 2
5875:| = 3
5689:which are
5452: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
8784:MathWorld
8767:EMS Press
8167:Citations
7948:metalogic
7655:-sets, a
7306:power set
6933:morphisms
6633:given by
6578:{1, ...,
6522:on a set
6060:∈
6053:∑
5779:⋅ 1
5533:⋅
5454:bijection
5430:over any
5334:∈
5327:⟺
5313:∈
5302:−
5293:⟺
5279:⋅
5267:−
5258:⟺
5250:⋅
5236:⋅
5228:⟺
5157: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
8818:Symmetry
8537:(2002),
8469:(2000).
8278:, p. 72.
8254:, p. 73.
8130:See also
8054:groupoid
7940:category
7863:acts on
7851:, where
7724:and all
7712:for all
7676: :
7657:morphism
7649:are two
7609:groupoid
7564:). This
7551:(where "
7507: :
7331: :
6946: :
6917:category
6801:acts on
6489:for all
6342:for all
6269:for all
6212:and all
6200:for all
6161:Examples
5736:Example:
5639:Example:
5517:, gives
5148: :
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
8769:, 2001
8743:0889050
8559:1867354
8489:1777008
8347:Algebra
8099:Gallery
8089:schemes
8081:actions
8038:to the
8028:to the
8020:functor
7964:monoids
7950:, this
7607:action
7386:versors
7384:1 (the
7304:on the
7144:−
6889:PGL(2,
6874:PGL(2,
6834:regular
6675:, ...,
6652:, ...,
6170:trivial
5495:⋅
5476:⋅
5441:⋅
5187:⋅
5166:⋅
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
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698:Lorentz
620:Unitary
519:Lattice
459:PSL(2,
193:abelian
104:(Semi-)
8741:
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3043:is by
2832:= 2, 3
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2016:(with
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553:Circle
484:SL(2,
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337:-group
188:cyclic
163:finite
158:simple
142:kernel
8162:Notes
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953:faces
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900:is a
891:on a
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147:image
8729:ISBN
8664:ISBN
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1076:left
1010:The
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823:The
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720:Loop
539:and
8721:doi
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31:.
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