5193:
3550:
action, such as a rotation around an axis or a reflection through a mirror plane. In other words, it is an operation that moves the molecule such that it is indistinguishable from the original configuration. In group theory, the rotation axes and mirror planes are called "symmetry elements". These elements can be a point, line or plane with respect to which the symmetry operation is carried out. The symmetry operations of a molecule determine the specific point group for this molecule.
5205:
2706:
827:
66:
5229:
5217:
3734:
3104:
3672:) many molecules have mirror planes, although they may not be obvious. The reflection operation exchanges left and right, as if each point had moved perpendicularly through the plane to a position exactly as far from the plane as when it started. When the plane is perpendicular to the principal axis of rotation, it is called
3549:
is responsible for many physical and spectroscopic properties of compounds and provides relevant information about how chemical reactions occur. In order to assign a point group for any given molecule, it is necessary to find the set of symmetry operations present on it. The symmetry operation is an
3711:
molecules lack inversion symmetry. To see this, hold a methane model with two hydrogen atoms in the vertical plane on the right and two hydrogen atoms in the horizontal plane on the left. Inversion results in two hydrogen atoms in the horizontal plane on the right and two hydrogen atoms in the
2945:
because if you take a symmetry of an object, and then apply another symmetry, the result will still be a symmetry. The identity keeping the object fixed is always a symmetry of an object. Existence of inverses is guaranteed by undoing the symmetry and the associativity comes from the fact that
2508:
studies groups from the perspective of generators and relations. It is particularly useful where finiteness assumptions are satisfied, for example finitely generated groups, or finitely presented groups (i.e. in addition the relations are finite). The area makes use of the connection of
3702:
Inversion (i ) is a more complex operation. Each point moves through the center of the molecule to a position opposite the original position and as far from the central point as where it started. Many molecules that seem at first glance to have an inversion center do not; for example,
3712:
vertical plane on the left. Inversion is therefore not a symmetry operation of methane, because the orientation of the molecule following the inversion operation differs from the original orientation. And the last operation is improper rotation or rotation reflection operation (
3322:
3554:
1563:, and so on. Rather than exploring properties of an individual group, one seeks to establish results that apply to a whole class of groups. The new paradigm was of paramount importance for the development of mathematics: it foreshadowed the creation of
940:
dates from the 19th century. One of the most important mathematical achievements of the 20th century was the collaborative effort, taking up more than 10,000 journal pages and mostly published between 1960 and 2004, that culminated in a complete
3461:
of the system. Physicists are very interested in group representations, especially of Lie groups, since these representations often point the way to the "possible" physical theories. Examples of the use of groups in physics include the
1088:
The different scope of these early sources resulted in different notions of groups. The theory of groups was unified starting around 1880. Since then, the impact of group theory has been ever growing, giving rise to the birth of
1415:
Most groups considered in the first stage of the development of group theory were "concrete", having been realized through numbers, permutations, or matrices. It was not until the late nineteenth century that the idea of an
1721:
2700:
4603:, the online mathematics magazine produced by the Millennium Mathematics Project at the University of Cambridge, exploring applications and recent breakthroughs, and giving explicit definitions and examples of groups.
3159:
have been introduced above. The presence of the group operation yields additional information which makes these varieties particularly accessible. They also often serve as a test for new conjectures. (For example the
4593:
This presents a view of group theory as level one of a theory that extends in all dimensions, and has applications in homotopy theory and to higher dimensional nonabelian methods for local-to-global problems.
3215:
2716:
attacks these problems from a geometric viewpoint, either by viewing groups as geometric objects, or by finding suitable geometric objects a group acts on. The first idea is made precise by means of the
3035:
cannot be solved by radicals in the way equations of lower degree can. The theory, being one of the historical roots of group theory, is still fruitfully applied to yield new results in areas such as
1757:
The presence of extra structure relates these types of groups with other mathematical disciplines and means that more tools are available in their study. Topological groups form a natural domain for
1543:
The change of perspective from concrete to abstract groups makes it natural to consider properties of groups that are independent of a particular realization, or in modern language, invariant under
2463:
3763:. Cryptographical methods of this kind benefit from the flexibility of the geometric objects, hence their group structures, together with the complicated structure of these groups, which make the
3220:
2014:
2595:
2496:
4284:
An introductory undergraduate text in the spirit of texts by
Gallian or Herstein, covering groups, rings, integral domains, fields and Galois theory. Free downloadable PDF with open-source
2991:(corresponding to addition) together with a second operation (corresponding to multiplication). Therefore, group theoretic arguments underlie large parts of the theory of those entities.
2396:
3599:) consists of leaving the molecule as it is. This is equivalent to any number of full rotations around any axis. This is a symmetry of all molecules, whereas the symmetry group of a
1476:
2345:
2104:, which is very explicit. On the other hand, given a well-understood group acting on a complicated object, this simplifies the study of the object in question. For example, if
2088:
This definition can be understood in two directions, both of which give rise to whole new domains of mathematics. On the one hand, it may yield new information about the group
1249:
In many cases, the structure of a permutation group can be studied using the properties of its action on the corresponding set. For example, in this way one proves that for
3866:. Thus Lie groups are group objects in the category of differentiable manifolds and affine algebraic groups are group objects in the category of affine algebraic varieties.
3409:
1332:. This action makes matrix groups conceptually similar to permutation groups, and the geometry of the action may be usefully exploited to establish properties of the group
2927:
511:
486:
449:
2900:
2876:
3603:
molecule consists of only the identity operation. An identity operation is a characteristic of every molecule even if it has no symmetry. Rotation around an axis (
2628:
3008:
uses groups to describe the symmetries of the roots of a polynomial (or more precisely the automorphisms of the algebras generated by these roots). The
1617:
2633:
3539:, circular dichroism spectroscopy, magnetic circular dichroism spectroscopy, UV/Vis spectroscopy, and fluorescence spectroscopy), and to construct
2223:
2721:, whose vertices correspond to group elements and edges correspond to right multiplication in the group. Given two elements, one constructs the
4653:
3016:
and group theory. It gives an effective criterion for the solvability of polynomial equations in terms of the solvability of the corresponding
813:
3978:
2933:
belonging to the equation. Every polynomial equation in one variable has a Galois group, that is a certain permutation group on its roots.
1773:. Certain classification questions that cannot be solved in general can be approached and resolved for special subclasses of groups. Thus,
1888:
of mathematical or physical objects, when those objects admit just a finite number of structure-preserving transformations. The theory of
4211:
4622:
3317:{\displaystyle {\begin{aligned}\sum _{n\geq 1}{\frac {1}{n^{s}}}&=\prod _{p{\text{ prime}}}{\frac {1}{1-p^{-s}}},\\\end{aligned}}\!}
1777:
have been completely classified. There is a fruitful relation between infinite abstract groups and topological groups: whenever a group
3829:
3080:, is a prominent application of this idea. The influence is not unidirectional, though. For example, algebraic topology makes use of
1841:
During the twentieth century, mathematicians investigated some aspects of the theory of finite groups in great depth, especially the
1045:
pushed these investigations further by creating the theory of permutation groups. The second historical source for groups stems from
2538:, which asks whether two groups given by different presentations are actually isomorphic. For example, the group with presentation
3976:
Abramovich, Dan; Karu, Kalle; Matsuki, Kenji; Wlodarczyk, Jaroslaw (2002), "Torification and factorization of birational maps",
3064:. They are called "invariants" because they are defined in such a way that they do not change if the space is subjected to some
2144:
1854:
1098:
942:
371:
4554:
4525:
4484:
4410:
4351:
4157:
4125:
4055:
3771:, may also be interpreted as a (very easy) group operation. Most cryptographic schemes use groups in some way. In particular
2401:
5016:
3960:
3009:
321:
1797:. A comparatively recent trend in the theory of finite groups exploits their connections with compact topological groups (
4149:
2839:
Symmetries are not restricted to geometrical objects, but include algebraic objects as well. For instance, the equation
1972:
5161:
4646:
4085:
806:
316:
3612:) consists of rotating the molecule around a specific axis by a specific angle. It is rotation through the angle 360°/
2541:
2468:
5233:
4502:
4454:
4438:
4376:
3401:
group and the concept of group action are often used to simplify the counting of a set of objects; see in particular
3328:
2143:
exist. There are several settings, and the employed methods and obtained results are rather different in every case:
3481:
Group theory can be used to resolve the incompleteness of the statistical interpretations of mechanics developed by
4697:
4628:
This is a detailed exposition of contemporaneous understanding of Group Theory by an early researcher in the field.
4590:
1735:
1122:
The range of groups being considered has gradually expanded from finite permutation groups and special examples of
4585:
2261:. An extension of Galois theory to the case of continuous symmetry groups was one of Lie's principal motivations.
894:
are two branches of group theory that have experienced advances and have become subject areas in their own right.
5111:
3772:
3453:, groups are important because they describe the symmetries which the laws of physics seem to obey. According to
5209:
4285:
3880:
3639:, since applying it twice produces the identity operation. In molecules with more than one rotation axis, the C
3168:
is studied in particular detail. They are both theoretically and practically intriguing. In another direction,
732:
2366:
1420:
began to take hold, where "abstract" means that the nature of the elements are ignored in such a way that two
886:. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra.
4639:
2778:
is a mapping of the object onto itself which preserves the structure. This occurs in many cases, for example
992:
799:
3060:
groups to the objects the theory is interested in. There, groups are used to describe certain invariants of
1438:
5221:
2953:
2510:
17:
866:. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as
5136:
4692:
4343:
4239:
3804:
3756:
3185:
3081:
2241:, which makes them indispensable tools for many parts of contemporary mathematics, as well as for modern
1782:
416:
230:
3643:
axis having the largest value of n is the highest order rotation axis or principal axis. For example in
2304:
4707:
4517:
3458:
2505:
2250:
2113:
1931:
1758:
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1008:
148:
5121:
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3013:
2535:
1131:
5166:
3780:
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614:
348:
225:
113:
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are two main subdomains of the theory. The totality of representations is governed by the group's
494:
469:
432:
5051:
5041:
5011:
4945:
4680:
4612:
4402:
4367:
The
Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry
3893:
3760:
3681:(horizontal). Other planes, which contain the principal axis of rotation, are labeled vertical (
3485:, relating to the summing of an infinite number of probabilities to yield a meaningful solution.
3438:
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Groups can be described in different ways. Finite groups can be described by writing down the
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4803:
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are considered as the same group. A typical way of specifying an abstract group is through a
1376:
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1042:
988:
914:
843:
638:
5156:
5036:
5031:
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4535:
4441:. For lay readers. Describes the quest to find the basic building blocks for finite groups.
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2020:
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184:
118:
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The saying of "preserving the structure" of an object can be made precise by working in a
2604:
2526:
asks whether two words are effectively the same group element. By relating the problem to
8:
5171:
5081:
5003:
4902:
4836:
4793:
4783:
4763:
3402:
3378:
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2520:
There are several natural questions arising from giving a group by its presentation. The
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4549:, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press,
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4061:
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4013:
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solving this task. Another, generally harder, algorithmically insoluble problem is the
2359:. The kernel of this map is called the subgroup of relations, generated by some subset
1963:
1147:
1066:
1050:
980:
964:
867:
138:
110:
3858:
This process of imposing extra structure has been formalized through the notion of a
1951:
has more structure, it is useful to restrict this notion further: a representation of
1030:
1000:
5192:
4912:
4897:
4841:
4788:
4568:
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4521:
4508:
Inexpensive and fairly readable, but somewhat dated in emphasis, style, and notation.
4498:
4480:
4450:
4434:
4416:
4406:
4372:
4347:
4311:
4223:
4153:
4121:
4081:
4051:
3788:
3644:
3540:
3524:
3520:
3504:
3370:
3364:
3173:
3093:
3069:
3061:
2517:. A fundamental theorem of this area is that every subgroup of a free group is free.
2514:
1900:. These are finite groups generated by reflections which act on a finite-dimensional
1774:
1743:
1600:
1592:
1540:; but the idea of an abstract group permits one not to worry about this discrepancy.
1421:
1257:
1151:
996:
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839:
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280:
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709:
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3523:. The assigned groups can then be used to determine physical properties (such as
3377:, that is, integrals invariant under the translation in a Lie group, are used for
2946:
symmetries are functions on a space, and composition of functions is associative.
2809:
of the set to itself which preserves the distance between each pair of points (an
1081:
problems. Thirdly, groups were, at first implicitly and later explicitly, used in
5066:
4993:
4922:
4715:
4560:
4531:
4476:
4327:
4261:
4242:; Stewart, Ian (2006), "Nonlinear dynamics of networks: the groupoid formalism",
4191:
4163:
4131:
4109:
4108:, Graduate Texts in Mathematics, vol. 126 (2nd ed.), Berlin, New York:
4039:
4005:
3156:
3085:
3028:
2164:
1901:
1873:
1850:
1798:
1751:
1502:
1364:. The concept of a transformation group is closely related with the concept of a
1271:
1207:
1135:
984:
831:
778:
771:
757:
714:
602:
525:
355:
269:
209:
89:
3725:, followed by reflection through a plane perpendicular to the axis of rotation.
1716:{\displaystyle m:G\times G\to G,(g,h)\mapsto gh,\quad i:G\to G,g\mapsto g^{-1},}
1375:
The theory of transformation groups forms a bridge connecting group theory with
5144:
5071:
4778:
4542:
4077:
3463:
3422:
3165:
2833:
2814:
2765:
2749:
2527:
2168:
2156:
1846:
1842:
1727:
1560:
1552:
1486:
1400:
1396:
1365:
1033:, in the 1830s, was the first to employ groups to determine the solvability of
972:
910:
906:
785:
721:
411:
391:
328:
293:
214:
204:
189:
174:
128:
105:
4117:
2956:. So every abstract group is actually the symmetries of some explicit object.
2695:{\displaystyle G\cong \langle z,y\mid z^{3}=y\rangle \cong \langle z\rangle .}
2245:. They provide a natural framework for analysing the continuous symmetries of
5249:
4932:
4864:
4816:
4617:
4394:
4315:
4227:
4143:
3834:
3768:
3565:, there are five important symmetry operations. They are identity operation (
3471:
3394:
3344:
3336:
3206:
3169:
3065:
3005:
3000:
2988:
2829:
2254:
2160:
1568:
1517:
1392:
1229:
1049:
situations. In an attempt to come to grips with possible geometries (such as
1038:
1004:
999:
in their quest for general solutions of polynomial equations of high degree.
960:
902:
704:
626:
460:
333:
199:
4626:, vol. 12 (11th ed.), Cambridge University Press, pp. 626–636
4572:
4080:, Cybernetics: Or Control and Communication in the Animal and the Machine,
835:
4874:
4869:
4468:
4382:
Conveys the practical value of group theory by explaining how it points to
4101:
3859:
3776:
3738:
3467:
3374:
3348:
3332:
3017:
2930:
2802:
2798:
2734:
2725:
given by the length of the minimal path between the elements. A theorem of
2718:
2040:
1956:
1904:. The properties of finite groups can thus play a role in subjects such as
1881:
1858:
1836:
1576:
1556:
1548:
1361:
1298:
1267:
1123:
1105:
875:
559:
258:
247:
194:
169:
164:
123:
94:
38:
4631:
4420:
4187:
4047:
3908:"Sur les invariants différentiels des groupes continus de transformations"
1481:
A significant source of abstract groups is given by the construction of a
5076:
4740:
4663:
4464:
4360:
4292:
Kleiner, Israel (1986), "The evolution of group theory: a brief survey",
3792:
3708:
3516:
3508:
3398:
3340:
2726:
2722:
2276:
1544:
1509:
1384:
1320:
that is closed under the products and inverses. Such a group acts on the
1062:
1020:
4611:
2498:
A string consisting of generator symbols and their inverses is called a
2120:). These parts, in turn, are much more easily manageable than the whole
1864:
During the second half of the twentieth century, mathematicians such as
5061:
4940:
4735:
4426:
4383:
4323:
3924:
2348:
2211:
2189:
2028:
1897:
1876:, and other related groups. One such family of groups is the family of
1731:
1572:
1516:
were among the earliest examples of factor groups, of much interest in
1380:
1102:
1070:
726:
454:
4038:, Lecture Notes in Computer Science, vol. 413, Berlin, New York:
2709:
The Cayley graph of ⟨ x, y ∣ ⟩, the free group of rank 2
2206:, with the property that the group operations are compatible with the
1352:
preserving its inherent structure. In the case of permutation groups,
1101:
is a vast body of work from the mid 20th century, classifying all the
3992:
3562:
3500:
2806:
2787:
2705:
2601:
of integers, although this may not be immediately apparent. (Writing
2531:
2195:
2148:
1909:
1889:
1762:
1747:
1163:
1074:
922:
891:
547:
4307:
3907:
3632:
atoms, it is in the same configuration as it started. In this case,
3441:
models musical transformations as elements of a mathematical group.
3072:"counts" how many paths in the space are essentially different. The
1765:(frequently realized as transformation groups) are the mainstays of
1027:
proclaimed group theory to be the organizing principle of geometry.
4965:
4884:
4811:
4599:
This package brings together all the articles on group theory from
4270:
3629:
3205:
makes uses of groups for some important applications. For example,
3100:
of an infinite group shows the legacy of topology in group theory.
2964:
2938:
2810:
2775:
2175:
1885:
1388:
1221:
1046:
1003:
coined the term "group" and established a connection, now known as
968:
84:
3767:
very hard to calculate. One of the earliest encryption protocols,
3457:, every continuous symmetry of a physical system corresponds to a
3412:
The circle of fifths may be endowed with a cyclic group structure.
4750:
4387:
3830:"An enormous theorem: the classification of finite simple groups"
3704:
3450:
3057:
1812:
918:
898:
826:
426:
340:
2222:
first appeared in French in 1893 in the thesis of Lie's student
1276:
impossibility of solving a general algebraic equation of degree
4174:
Cannon, John J. (1969), "Computers in group theory: A survey",
3975:
3625:
3600:
65:
3624:
molecule rotates 180° around the axis that passes through the
3031:
in 5 elements, is not solvable which implies that the general
2979:
Applications of group theory abound. Almost all structures in
2929:. In this case, the group that exchanges the two roots is the
2790:
map from the set to itself, giving rise to permutation groups.
1547:, as well as the classes of group with a given such property:
1232:
exhibited any group as a permutation group, acting on itself (
3621:
3177:
3107:
A torus. Its abelian group structure is induced from the map
2825:
1587:
An important elaboration of the concept of a group occurs if
909:
known fundamental forces in the universe, may be modelled by
883:
4212:"Herstellung von Graphen mit vorgegebener abstrakter Gruppe"
30:
This article covers advanced notions. For basic topics, see
2264:
27:
Branch of mathematics that studies the properties of groups
4278:
3733:
3586:) and rotation reflection operation or improper rotation (
3369:
Analysis on Lie groups and certain other groups is called
2253:), in much the same way as permutation groups are used in
1344:
Permutation groups and matrix groups are special cases of
987:. Early results about permutation groups were obtained by
3620:
is an integer, about a rotation axis. For example, if a
3408:
2937:
The axioms of a group formalize the essential aspects of
1861:
from which all finite groups can be built are now known.
3553:
3103:
3180:. Toroidal embeddings have recently led to advances in
2786:
is a set with no additional structure, a symmetry is a
2214:, who laid the foundations of the theory of continuous
1822:
translate into the properties of its finite quotients.
3164:(in certain cases).) The one-dimensional case, namely
2458:{\displaystyle \langle a,b\mid aba^{-1}b^{-1}\rangle }
1872:
also increased our understanding of finite analogs of
1726:
are compatible with this structure, that is, they are
3531:), spectroscopic properties (particularly useful for
3218:
2908:
2884:
2845:
2636:
2607:
2544:
2471:
2404:
2369:
2307:
1975:
1620:
1441:
497:
472:
435:
2952:
says that every group is the symmetry group of some
2759:
1591:
is endowed with additional structure, notably, of a
878:, can all be seen as groups endowed with additional
1947:in a way compatible with the group structure. When
1582:
1372:transformations that preserve a certain structure.
1126:to abstract groups that may be specified through a
4364:
4269:Shows the advantage of generalising from group to
4031:
3488:
3316:
2921:
2894:
2870:
2752:(i.e. looks similar from a distance) to the space
2694:
2622:
2589:
2490:
2457:
2390:
2339:
2229:Lie groups represent the best-developed theory of
2009:{\displaystyle \rho :G\to \operatorname {GL} (V),}
2008:
1715:
1470:
1174:) that is closed under compositions and inverses,
1097:, and many more influential spin-off domains. The
983:and additive and multiplicative groups related to
505:
480:
443:
4238:
3331:that any integer decomposes in a unique way into
3313:
5247:
3755:Very large groups of prime order constructed in
3511:are used to classify regular polyhedra, and the
2590:{\displaystyle \langle x,y\mid xyxyx=e\rangle ,}
2491:{\displaystyle \mathbb {Z} \times \mathbb {Z} .}
1738:(in the sense of algebraic geometry) maps, then
1011:. In geometry, groups first became important in
959:Group theory has three main historical sources:
2974:
2289:. A more compact way of defining a group is by
1884:. Finite groups often occur when considering
1293:The next important class of groups is given by
4597:Plus teacher and student package: Group Theory
4444:
1368:: transformation groups frequently consist of
4647:
4034:Group theoretical methods in image processing
3905:
2963:. Maps preserving the structure are then the
807:
4492:
4463:
3979:Journal of the American Mathematical Society
3937:
2686:
2680:
2674:
2643:
2581:
2545:
2452:
2405:
2382:
2370:
2322:
2308:
1896:", is strongly influenced by the associated
1462:
1448:
1379:. A long line of research, originating with
1150:of groups to undergo a systematic study was
1073:, in 1884, started using groups (now called
913:. Thus group theory and the closely related
4661:
2530:, one can show that there is in general no
2279:consisting of all possible multiplications
1915:
1825:
1811:has a family of quotients which are finite
1007:, between the nascent theory of groups and
971:. The number-theoretic strand was begun by
4654:
4640:
4514:Profinite groups, arithmetic, and geometry
3191:
1789:, the geometry and analysis pertaining to
814:
800:
4359:
4280:Abstract Algebra: Theory and Applications
4255:
3991:
3923:
3651:), the highest order of rotation axis is
3569:, rotation operation or proper rotation (
3155:likewise uses group theory in many ways.
2797:is a set of points in the plane with its
2481:
2473:
2465:describes a group which is isomorphic to
2363:. The presentation is usually denoted by
2257:for analysing the discrete symmetries of
2035:. In other words, to every group element
1111:
499:
474:
437:
37:For group theory in social sciences, see
4606:
4337:
3948:
3892:In particular, if the representation is
3732:
3658:, so the principal axis of rotation is
3552:
3407:
3102:
2704:
2391:{\displaystyle \langle F\mid D\rangle .}
2265:Combinatorial and geometric group theory
2159:can be interpreted as the characters of
1339:
825:
4447:An introduction to the theory of groups
4393:
4291:
4148:, Classroom Resource Materials Series,
2729:and Svarc then says that given a group
1892:, which may be viewed as dealing with "
14:
5248:
4547:An introduction to homological algebra
4541:
4276:
4209:
4173:
4141:
2145:representation theory of finite groups
1855:classification of finite simple groups
1830:
1579:, and mathematicians of their school.
1471:{\displaystyle G=\langle S|R\rangle .}
1099:classification of finite simple groups
943:classification of finite simple groups
838:, has been used as an illustration of
372:Classification of finite simple groups
4635:
4586:History of the abstract group concept
4511:
4100:
3827:
3141:
3042:
2813:). The corresponding group is called
1857:was achieved, meaning that all those
1818:of various orders, and properties of
1348:: groups that act on a certain space
1141:
5216:
4029:
3961:Birch and Swinnerton-Dyer conjecture
3358:
3335:. The failure of this statement for
3096:of groups. Finally, the name of the
3056:is another domain which prominently
3010:fundamental theorem of Galois theory
2941:. Symmetries form a group: they are
2597:is isomorphic to the additive group
2398:For example, the group presentation
1274:. This fact plays a key role in the
1216:; in general, any permutation group
917:have many important applications in
5228:
4150:Mathematical Association of America
2733:acting in a reasonable manner on a
1943:defines a bijective map on the set
1845:of finite groups and the theory of
24:
4460:A standard contemporary reference.
3721:) requires rotation of 360°/
2340:{\displaystyle \{g_{i}\}_{i\in I}}
2139:then asks what representations of
1410:
1305:is a set consisting of invertible
929:. Group theory is also central to
897:Various physical systems, such as
25:
5267:
4579:
3557:Water molecule with symmetry axis
3084:which are spaces with prescribed
2760:Connection of groups and symmetry
1853:. As a consequence, the complete
1270:, i.e. does not admit any proper
5227:
5215:
5204:
5203:
5191:
4340:Topics in geometric group theory
3828:Elwes, Richard (December 2006),
3388:
2994:
2987:, for example, can be viewed as
2967:, and the symmetry group is the
2092:: often, the group operation in
1583:Groups with additional structure
1524:is a permutation group on a set
1288:
64:
5112:Computational complexity theory
4591:Higher dimensional group theory
4071:
4023:
3728:
3489:Chemistry and materials science
1672:
1399:. The groups themselves may be
1228:. An early construction due to
3969:
3953:
3942:
3931:
3899:
3886:
3869:
3852:
3821:
2832:. Conformal maps give rise to
2180:-space of periodic functions.
2000:
1994:
1985:
1793:yield important results about
1694:
1682:
1660:
1657:
1645:
1636:
1455:
733:Infinite dimensional Lie group
13:
1:
4449:, New York: Springer-Verlag,
4257:10.1090/S0273-0979-06-01108-6
4244:Bull. Amer. Math. Soc. (N.S.)
4094:
4002:10.1090/S0894-0347-02-00396-X
3668:In the reflection operation (
3133:is a parameter living in the
2983:are special cases of groups.
2828:are preserved, one speaks of
2210:. Lie groups are named after
2183:
2096:is abstractly given, but via
2027:) consists of the invertible
1387:, considers group actions on
1356:is a set; for matrix groups,
4338:La Harpe, Pierre de (2000),
2975:Applications of group theory
2922:{\displaystyle -{\sqrt {3}}}
2108:is finite, it is known that
1939:means that every element of
1775:compact connected Lie groups
834:puzzle, invented in 1974 by
506:{\displaystyle \mathbb {Z} }
481:{\displaystyle \mathbb {Z} }
444:{\displaystyle \mathbb {Z} }
7:
4433:. Oxford University Press.
4344:University of Chicago Press
3805:List of group theory topics
3798:
3773:Diffie–Hellman key exchange
3757:elliptic curve cryptography
3595:). The identity operation (
3186:resolution of singularities
2971:of the object in question.
2895:{\displaystyle {\sqrt {3}}}
1093:in the early 20th century,
231:List of group theory topics
10:
5272:
5162:Films about mathematicians
4518:Princeton University Press
4512:Shatz, Stephen S. (1972),
4473:Combinatorial group theory
4277:Judson, Thomas W. (1997),
4142:Carter, Nathan C. (2009),
3492:
3444:
3362:
3195:
3145:
3046:
3014:algebraic field extensions
2998:
2770:Given a structured object
2763:
2506:Combinatorial group theory
2297:of a group. Given any set
2268:
2251:differential Galois theory
2187:
2102:multiplication of matrices
1919:
1834:
1759:abstract harmonic analysis
1603:. If the group operations
1324:-dimensional vector space
1224:of the symmetric group of
1115:
952:
948:
36:
29:
5185:
5135:
5092:
5002:
4964:
4931:
4883:
4855:
4802:
4749:
4731:Philosophy of mathematics
4706:
4671:
4613:"Groups, Theory of"
4176:Communications of the ACM
4118:10.1007/978-1-4612-0941-6
3578:), reflection operation (
3076:, proved in 2002/2003 by
2871:{\displaystyle x^{2}-3=0}
2536:group isomorphism problem
1801:): for example, a single
5167:Recreational mathematics
4431:Symmetry and the Monster
4371:, Simon & Schuster,
3938:Schupp & Lyndon 2001
3815:
3781:group-based cryptography
3416:
3082:Eilenberg–MacLane spaces
3012:provides a link between
2355:surjects onto the group
2291:generators and relations
2100:, it corresponds to the
1916:Representation of groups
1826:Branches of group theory
1430:generators and relations
349:Elementary abelian group
226:Glossary of group theory
5052:Mathematical statistics
5042:Mathematical psychology
5012:Engineering mathematics
4946:Algebraic number theory
4623:Encyclopædia Britannica
4445:Rotman, Joseph (1994),
4403:Oxford University Press
4106:Linear algebraic groups
3785:cryptographic protocols
3761:public-key cryptography
3513:symmetries of molecules
3439:Transformational theory
3431:elementary group theory
3429:yields applications of
3421:The presence of the 12-
3207:Euler's product formula
3203:Algebraic number theory
3198:Algebraic number theory
3192:Algebraic number theory
2801:structure or any other
2204:differentiable manifold
2147:and representations of
1785:in a topological group
1597:differentiable manifold
1536:is no longer acting on
1514:algebraic number fields
1242:) by means of the left
1083:algebraic number theory
955:History of group theory
938:history of group theory
931:public key cryptography
888:Linear algebraic groups
5198:Mathematics portal
5047:Mathematical sociology
5027:Mathematical economics
5022:Mathematical chemistry
4951:Analytic number theory
4832:Differential equations
4493:Scott, W. R. (1987) ,
4216:Compositio Mathematica
3906:Arthur Tresse (1893),
3752:
3558:
3413:
3318:
3138:
2923:
2896:
2878:has the two solutions
2872:
2714:Geometric group theory
2710:
2696:
2624:
2591:
2492:
2459:
2392:
2341:
2271:Geometric group theory
2247:differential equations
2112:above decomposes into
2029:linear transformations
2010:
1717:
1472:
1330:linear transformations
1244:regular representation
1170:into itself (known as
1112:Main classes of groups
1061:) using group theory,
1017:non-Euclidean geometry
847:
765:Linear algebraic group
507:
482:
445:
5177:Mathematics education
5107:Theory of computation
4827:Hypercomplex analysis
4188:10.1145/362835.362837
4048:10.1007/3-540-52290-5
4030:Lenz, Reiner (1990),
3736:
3628:atom and between the
3556:
3537:infrared spectroscopy
3411:
3353:Fermat's Last Theorem
3319:
3106:
2924:
2897:
2873:
2708:
2697:
2625:
2592:
2493:
2460:
2393:
2342:
2216:transformation groups
2137:representation theory
2011:
1922:Representation theory
1878:general linear groups
1781:can be realized as a
1771:representation theory
1767:differential geometry
1718:
1607:(multiplication) and
1473:
1377:differential geometry
1346:transformation groups
1340:Transformation groups
1095:representation theory
1043:Augustin Louis Cauchy
915:representation theory
829:
508:
483:
446:
5157:Informal mathematics
5037:Mathematical physics
5032:Mathematical finance
5017:Mathematical biology
4956:Diophantine geometry
4475:, Berlin, New York:
4295:Mathematics Magazine
3881:equivariant K-theory
3838:(41), archived from
3216:
2906:
2882:
2843:
2634:
2623:{\displaystyle z=xy}
2605:
2542:
2469:
2402:
2367:
2305:
2235:mathematical objects
1973:
1926:Saying that a group
1806:-adic analytic group
1618:
1439:
1035:polynomial equations
860:algebraic structures
495:
470:
433:
5172:Mathematics and art
5082:Operations research
4837:Functional analysis
4497:, New York: Dover,
4390:and other sciences.
4210:Frucht, R. (1939),
4145:Visual group theory
3965:millennium problems
3379:pattern recognition
3347:, which feature in
3174:algebraic varieties
3092:relies in a way on
3074:Poincaré conjecture
3068:. For example, the
2259:algebraic equations
2243:theoretical physics
2231:continuous symmetry
2157:Fourier polynomials
1906:theoretical physics
1894:continuous symmetry
1831:Finite group theory
1528:, the factor group
1118:Group (mathematics)
1059:projective geometry
1013:projective geometry
975:, and developed by
965:algebraic equations
139:Group homomorphisms
49:Algebraic structure
32:Group (mathematics)
5117:Numerical analysis
4726:Mathematical logic
4721:Information theory
4543:Weibel, Charles A.
4240:Golubitsky, Martin
3925:10.1007/bf02418270
3810:Examples of groups
3789:non-abelian groups
3787:that use infinite
3765:discrete logarithm
3753:
3559:
3547:Molecular symmetry
3541:molecular orbitals
3533:Raman spectroscopy
3521:crystal structures
3495:Molecular symmetry
3435:musical set theory
3414:
3337:more general rings
3314:
3311:
3277:
3238:
3182:algebraic geometry
3153:Algebraic geometry
3148:Algebraic geometry
3142:Algebraic geometry
3139:
3094:classifying spaces
3090:algebraic K-theory
3062:topological spaces
3054:Algebraic topology
3049:Algebraic topology
3043:Algebraic topology
3037:class field theory
2969:automorphism group
2919:
2892:
2868:
2805:, a symmetry is a
2711:
2692:
2620:
2587:
2515:fundamental groups
2488:
2455:
2388:
2337:
2293:, also called the
2006:
1964:group homomorphism
1713:
1468:
1152:permutation groups
1142:Permutation groups
1067:Erlangen programme
981:modular arithmetic
848:
844:Rubik's Cube group
840:permutation groups
615:Special orthogonal
503:
478:
441:
322:Lagrange's theorem
5243:
5242:
4842:Harmonic analysis
4608:Burnside, William
4556:978-0-521-55987-4
4527:978-0-691-08017-8
4486:978-3-540-41158-1
4412:978-0-19-560528-0
4399:Abelian varieties
4353:978-0-226-31721-2
4159:978-0-88385-757-1
4127:978-0-387-97370-8
4057:978-0-387-52290-6
3783:refers mostly to
3645:boron trifluoride
3525:chemical polarity
3505:materials science
3455:Noether's theorem
3371:harmonic analysis
3365:Harmonic analysis
3359:Harmonic analysis
3304:
3274:
3263:
3254:
3223:
3157:Abelian varieties
3070:fundamental group
2917:
2890:
2118:Maschke's theorem
2114:irreducible parts
1744:topological group
1601:algebraic variety
1593:topological space
1422:isomorphic groups
1258:alternating group
1158:and a collection
927:materials science
907:three of the four
824:
823:
399:
398:
281:Alternating group
238:
237:
16:(Redirected from
5263:
5231:
5230:
5219:
5218:
5207:
5206:
5196:
5195:
5127:Computer algebra
5102:Computer science
4822:Complex analysis
4656:
4649:
4642:
4633:
4632:
4627:
4615:
4575:
4538:
4507:
4489:
4469:Lyndon, Roger C.
4459:
4423:
4381:
4370:
4356:
4334:
4283:
4268:
4259:
4235:
4230:, archived from
4206:
4170:
4138:
4089:
4075:
4069:
4068:
4037:
4027:
4021:
4020:
3995:
3973:
3967:
3957:
3951:
3946:
3940:
3935:
3929:
3928:
3927:
3912:Acta Mathematica
3903:
3897:
3890:
3884:
3877:group cohomology
3873:
3867:
3856:
3850:
3849:
3848:
3847:
3825:
3638:
3459:conservation law
3427:circle of fifths
3403:Burnside's lemma
3397:, the notion of
3383:image processing
3323:
3321:
3320:
3315:
3312:
3305:
3303:
3302:
3301:
3279:
3276:
3275:
3272:
3255:
3253:
3252:
3240:
3237:
3184:, in particular
3162:Hodge conjecture
3135:upper half plane
3128:
3098:torsion subgroup
3078:Grigori Perelman
3033:quintic equation
2981:abstract algebra
2950:Frucht's theorem
2928:
2926:
2925:
2920:
2918:
2913:
2901:
2899:
2898:
2893:
2891:
2886:
2877:
2875:
2874:
2869:
2855:
2854:
2742:compact manifold
2740:, for example a
2701:
2699:
2698:
2693:
2667:
2666:
2629:
2627:
2626:
2621:
2596:
2594:
2593:
2588:
2497:
2495:
2494:
2489:
2484:
2476:
2464:
2462:
2461:
2456:
2451:
2450:
2438:
2437:
2397:
2395:
2394:
2389:
2346:
2344:
2343:
2338:
2336:
2335:
2320:
2319:
2288:
2208:smooth structure
2174:, acting on the
2076:
2015:
2013:
2012:
2007:
1874:classical groups
1851:nilpotent groups
1799:profinite groups
1722:
1720:
1719:
1714:
1709:
1708:
1567:in the works of
1565:abstract algebra
1477:
1475:
1474:
1469:
1458:
1282:
1272:normal subgroups
1255:
1241:
1154:. Given any set
1091:abstract algebra
1025:Erlangen program
985:quadratic fields
963:, the theory of
852:abstract algebra
816:
809:
802:
758:Algebraic groups
531:Hyperbolic group
521:Arithmetic group
512:
510:
509:
504:
502:
487:
485:
484:
479:
477:
450:
448:
447:
442:
440:
363:Schur multiplier
317:Cauchy's theorem
305:Quaternion group
253:
252:
79:
78:
68:
55:
44:
43:
21:
5271:
5270:
5266:
5265:
5264:
5262:
5261:
5260:
5246:
5245:
5244:
5239:
5190:
5181:
5131:
5088:
5067:Systems science
4998:
4994:Homotopy theory
4960:
4927:
4879:
4851:
4798:
4745:
4716:Category theory
4702:
4667:
4660:
4582:
4557:
4528:
4505:
4487:
4477:Springer-Verlag
4465:Schupp, Paul E.
4457:
4413:
4379:
4354:
4308:10.2307/2690312
4160:
4128:
4110:Springer-Verlag
4097:
4092:
4076:
4072:
4058:
4040:Springer-Verlag
4028:
4024:
3974:
3970:
3958:
3954:
3947:
3943:
3936:
3932:
3904:
3900:
3891:
3887:
3874:
3870:
3857:
3853:
3845:
3843:
3826:
3822:
3818:
3801:
3769:Caesar's cipher
3749:Caesar's cipher
3746:
3731:
3719:
3697:
3690:) or dihedral (
3688:
3679:
3663:
3656:
3650:
3642:
3633:
3610:
3593:
3576:
3497:
3491:
3447:
3419:
3391:
3367:
3361:
3310:
3309:
3294:
3290:
3283:
3278:
3271:
3267:
3256:
3248:
3244:
3239:
3227:
3219:
3217:
3214:
3213:
3200:
3194:
3170:toric varieties
3166:elliptic curves
3150:
3144:
3108:
3086:homotopy groups
3051:
3045:
3029:symmetric group
3026:
3020:. For example,
3003:
2997:
2977:
2912:
2907:
2904:
2903:
2885:
2883:
2880:
2879:
2850:
2846:
2844:
2841:
2840:
2834:Kleinian groups
2774:of any sort, a
2768:
2762:
2750:quasi-isometric
2662:
2658:
2635:
2632:
2631:
2606:
2603:
2602:
2543:
2540:
2539:
2528:Turing machines
2480:
2472:
2470:
2467:
2466:
2443:
2439:
2430:
2426:
2403:
2400:
2399:
2368:
2365:
2364:
2325:
2321:
2315:
2311:
2306:
2303:
2302:
2280:
2273:
2267:
2202:that is also a
2192:
2186:
2165:complex numbers
2163:, the group of
2155:. For example,
2051:
2039:is assigned an
1974:
1971:
1970:
1924:
1918:
1902:Euclidean space
1839:
1833:
1828:
1752:algebraic group
1701:
1697:
1619:
1616:
1615:
1585:
1561:solvable groups
1553:periodic groups
1503:normal subgroup
1454:
1440:
1437:
1436:
1413:
1411:Abstract groups
1397:diffeomorphisms
1342:
1309:of given order
1291:
1277:
1265:
1250:
1233:
1215:
1208:symmetric group
1144:
1120:
1114:
1001:Évariste Galois
957:
951:
911:symmetry groups
820:
791:
790:
779:Abelian variety
772:Reductive group
760:
750:
749:
748:
747:
698:
690:
682:
674:
666:
639:Special unitary
550:
536:
535:
517:
516:
498:
496:
493:
492:
473:
471:
468:
467:
436:
434:
431:
430:
422:
421:
412:Discrete groups
401:
400:
356:Frobenius group
301:
288:
277:
270:Symmetric group
266:
250:
240:
239:
90:Normal subgroup
76:
56:
47:
42:
35:
28:
23:
22:
15:
12:
11:
5:
5269:
5259:
5258:
5241:
5240:
5238:
5237:
5225:
5213:
5201:
5186:
5183:
5182:
5180:
5179:
5174:
5169:
5164:
5159:
5154:
5153:
5152:
5145:Mathematicians
5141:
5139:
5137:Related topics
5133:
5132:
5130:
5129:
5124:
5119:
5114:
5109:
5104:
5098:
5096:
5090:
5089:
5087:
5086:
5085:
5084:
5079:
5074:
5072:Control theory
5064:
5059:
5054:
5049:
5044:
5039:
5034:
5029:
5024:
5019:
5014:
5008:
5006:
5000:
4999:
4997:
4996:
4991:
4986:
4981:
4976:
4970:
4968:
4962:
4961:
4959:
4958:
4953:
4948:
4943:
4937:
4935:
4929:
4928:
4926:
4925:
4920:
4915:
4910:
4905:
4900:
4895:
4889:
4887:
4881:
4880:
4878:
4877:
4872:
4867:
4861:
4859:
4853:
4852:
4850:
4849:
4847:Measure theory
4844:
4839:
4834:
4829:
4824:
4819:
4814:
4808:
4806:
4800:
4799:
4797:
4796:
4791:
4786:
4781:
4776:
4771:
4766:
4761:
4755:
4753:
4747:
4746:
4744:
4743:
4738:
4733:
4728:
4723:
4718:
4712:
4710:
4704:
4703:
4701:
4700:
4695:
4690:
4689:
4688:
4683:
4672:
4669:
4668:
4659:
4658:
4651:
4644:
4636:
4630:
4629:
4618:Chisholm, Hugh
4604:
4594:
4588:
4581:
4580:External links
4578:
4577:
4576:
4555:
4539:
4526:
4509:
4503:
4490:
4485:
4461:
4455:
4442:
4424:
4411:
4395:Mumford, David
4391:
4377:
4357:
4352:
4335:
4302:(4): 195–215,
4289:
4274:
4250:(3): 305–364,
4236:
4207:
4171:
4158:
4139:
4126:
4096:
4093:
4091:
4090:
4086:978-0262730099
4078:Norbert Wiener
4070:
4056:
4022:
3986:(3): 531–572,
3968:
3952:
3941:
3930:
3898:
3885:
3868:
3862:in a suitable
3851:
3819:
3817:
3814:
3813:
3812:
3807:
3800:
3797:
3779:. So the term
3744:
3730:
3727:
3715:
3693:
3684:
3675:
3661:
3654:
3648:
3640:
3606:
3589:
3582:), inversion (
3572:
3493:Main article:
3490:
3487:
3476:Poincaré group
3464:Standard Model
3446:
3443:
3418:
3415:
3390:
3387:
3363:Main article:
3360:
3357:
3345:regular primes
3339:gives rise to
3325:
3324:
3308:
3300:
3297:
3293:
3289:
3286:
3282:
3270:
3266:
3262:
3259:
3257:
3251:
3247:
3243:
3236:
3233:
3230:
3226:
3222:
3221:
3196:Main article:
3193:
3190:
3176:acted on by a
3146:Main article:
3143:
3140:
3047:Main article:
3044:
3041:
3024:
2999:Main article:
2996:
2993:
2989:abelian groups
2976:
2973:
2935:
2934:
2916:
2911:
2889:
2867:
2864:
2861:
2858:
2853:
2849:
2837:
2836:, for example.
2830:conformal maps
2822:
2815:isometry group
2793:If the object
2791:
2766:Symmetry group
2764:Main article:
2761:
2758:
2691:
2688:
2685:
2682:
2679:
2676:
2673:
2670:
2665:
2661:
2657:
2654:
2651:
2648:
2645:
2642:
2639:
2619:
2616:
2613:
2610:
2586:
2583:
2580:
2577:
2574:
2571:
2568:
2565:
2562:
2559:
2556:
2553:
2550:
2547:
2487:
2483:
2479:
2475:
2454:
2449:
2446:
2442:
2436:
2433:
2429:
2425:
2422:
2419:
2416:
2413:
2410:
2407:
2387:
2384:
2381:
2378:
2375:
2372:
2334:
2331:
2328:
2324:
2318:
2314:
2310:
2301:of generators
2269:Main article:
2266:
2263:
2220:groupes de Lie
2188:Main article:
2185:
2182:
2169:absolute value
2131:Given a group
2017:
2016:
2005:
2002:
1999:
1996:
1993:
1990:
1987:
1984:
1981:
1978:
1920:Main article:
1917:
1914:
1835:Main article:
1832:
1829:
1827:
1824:
1724:
1723:
1712:
1707:
1704:
1700:
1696:
1693:
1690:
1687:
1684:
1681:
1678:
1675:
1671:
1668:
1665:
1662:
1659:
1656:
1653:
1650:
1647:
1644:
1641:
1638:
1635:
1632:
1629:
1626:
1623:
1584:
1581:
1487:quotient group
1479:
1478:
1467:
1464:
1461:
1457:
1453:
1450:
1447:
1444:
1418:abstract group
1412:
1409:
1393:homeomorphisms
1366:symmetry group
1341:
1338:
1290:
1287:
1261:
1211:
1202:permutations,
1143:
1140:
1116:Main article:
1113:
1110:
1077:) attached to
1065:initiated the
973:Leonhard Euler
953:Main article:
950:
947:
822:
821:
819:
818:
811:
804:
796:
793:
792:
789:
788:
786:Elliptic curve
782:
781:
775:
774:
768:
767:
761:
756:
755:
752:
751:
746:
745:
742:
739:
735:
731:
730:
729:
724:
722:Diffeomorphism
718:
717:
712:
707:
701:
700:
696:
692:
688:
684:
680:
676:
672:
668:
664:
659:
658:
647:
646:
635:
634:
623:
622:
611:
610:
599:
598:
587:
586:
579:Special linear
575:
574:
567:General linear
563:
562:
557:
551:
542:
541:
538:
537:
534:
533:
528:
523:
515:
514:
501:
489:
476:
463:
461:Modular groups
459:
458:
457:
452:
439:
423:
420:
419:
414:
408:
407:
406:
403:
402:
397:
396:
395:
394:
389:
384:
381:
375:
374:
368:
367:
366:
365:
359:
358:
352:
351:
346:
337:
336:
334:Hall's theorem
331:
329:Sylow theorems
325:
324:
319:
311:
310:
309:
308:
302:
297:
294:Dihedral group
290:
289:
284:
278:
273:
267:
262:
251:
246:
245:
242:
241:
236:
235:
234:
233:
228:
220:
219:
218:
217:
212:
207:
202:
197:
192:
187:
185:multiplicative
182:
177:
172:
167:
159:
158:
157:
156:
151:
143:
142:
134:
133:
132:
131:
129:Wreath product
126:
121:
116:
114:direct product
108:
106:Quotient group
100:
99:
98:
97:
92:
87:
77:
74:
73:
70:
69:
61:
60:
26:
9:
6:
4:
3:
2:
5268:
5257:
5254:
5253:
5251:
5236:
5235:
5226:
5224:
5223:
5214:
5212:
5211:
5202:
5200:
5199:
5194:
5188:
5187:
5184:
5178:
5175:
5173:
5170:
5168:
5165:
5163:
5160:
5158:
5155:
5151:
5148:
5147:
5146:
5143:
5142:
5140:
5138:
5134:
5128:
5125:
5123:
5120:
5118:
5115:
5113:
5110:
5108:
5105:
5103:
5100:
5099:
5097:
5095:
5094:Computational
5091:
5083:
5080:
5078:
5075:
5073:
5070:
5069:
5068:
5065:
5063:
5060:
5058:
5055:
5053:
5050:
5048:
5045:
5043:
5040:
5038:
5035:
5033:
5030:
5028:
5025:
5023:
5020:
5018:
5015:
5013:
5010:
5009:
5007:
5005:
5001:
4995:
4992:
4990:
4987:
4985:
4982:
4980:
4977:
4975:
4972:
4971:
4969:
4967:
4963:
4957:
4954:
4952:
4949:
4947:
4944:
4942:
4939:
4938:
4936:
4934:
4933:Number theory
4930:
4924:
4921:
4919:
4916:
4914:
4911:
4909:
4906:
4904:
4901:
4899:
4896:
4894:
4891:
4890:
4888:
4886:
4882:
4876:
4873:
4871:
4868:
4866:
4865:Combinatorics
4863:
4862:
4860:
4858:
4854:
4848:
4845:
4843:
4840:
4838:
4835:
4833:
4830:
4828:
4825:
4823:
4820:
4818:
4817:Real analysis
4815:
4813:
4810:
4809:
4807:
4805:
4801:
4795:
4792:
4790:
4787:
4785:
4782:
4780:
4777:
4775:
4772:
4770:
4767:
4765:
4762:
4760:
4757:
4756:
4754:
4752:
4748:
4742:
4739:
4737:
4734:
4732:
4729:
4727:
4724:
4722:
4719:
4717:
4714:
4713:
4711:
4709:
4705:
4699:
4696:
4694:
4691:
4687:
4684:
4682:
4679:
4678:
4677:
4674:
4673:
4670:
4665:
4657:
4652:
4650:
4645:
4643:
4638:
4637:
4634:
4625:
4624:
4619:
4614:
4609:
4605:
4602:
4598:
4595:
4592:
4589:
4587:
4584:
4583:
4574:
4570:
4566:
4562:
4558:
4552:
4548:
4544:
4540:
4537:
4533:
4529:
4523:
4519:
4515:
4510:
4506:
4504:0-486-65377-3
4500:
4496:
4491:
4488:
4482:
4478:
4474:
4470:
4466:
4462:
4458:
4456:0-387-94285-8
4452:
4448:
4443:
4440:
4439:0-19-280722-6
4436:
4432:
4428:
4425:
4422:
4418:
4414:
4408:
4404:
4400:
4396:
4392:
4389:
4385:
4380:
4378:0-7432-5820-7
4374:
4369:
4368:
4362:
4358:
4355:
4349:
4345:
4341:
4336:
4333:
4329:
4325:
4321:
4317:
4313:
4309:
4305:
4301:
4297:
4296:
4290:
4287:
4282:
4281:
4275:
4272:
4267:
4263:
4258:
4253:
4249:
4245:
4241:
4237:
4234:on 2008-12-01
4233:
4229:
4225:
4221:
4217:
4213:
4208:
4205:
4201:
4197:
4193:
4189:
4185:
4181:
4177:
4172:
4169:
4165:
4161:
4155:
4151:
4147:
4146:
4140:
4137:
4133:
4129:
4123:
4119:
4115:
4111:
4107:
4103:
4102:Borel, Armand
4099:
4098:
4087:
4083:
4079:
4074:
4067:
4063:
4059:
4053:
4049:
4045:
4041:
4036:
4035:
4026:
4019:
4015:
4011:
4007:
4003:
3999:
3994:
3989:
3985:
3981:
3980:
3972:
3966:
3963:, one of the
3962:
3956:
3950:
3949:La Harpe 2000
3945:
3939:
3934:
3926:
3921:
3917:
3913:
3909:
3902:
3895:
3889:
3882:
3878:
3872:
3865:
3861:
3855:
3842:on 2009-02-02
3841:
3837:
3836:
3835:Plus Magazine
3831:
3824:
3820:
3811:
3808:
3806:
3803:
3802:
3796:
3794:
3790:
3786:
3782:
3778:
3777:cyclic groups
3774:
3770:
3766:
3762:
3758:
3750:
3743:
3740:
3735:
3726:
3724:
3720:
3718:
3710:
3706:
3700:
3698:
3696:
3689:
3687:
3680:
3678:
3671:
3666:
3664:
3657:
3646:
3636:
3631:
3627:
3623:
3619:
3615:
3611:
3609:
3602:
3598:
3594:
3592:
3585:
3581:
3577:
3575:
3568:
3564:
3555:
3551:
3548:
3544:
3542:
3538:
3534:
3530:
3526:
3522:
3518:
3514:
3510:
3506:
3502:
3496:
3486:
3484:
3483:Willard Gibbs
3479:
3477:
3473:
3472:Lorentz group
3469:
3465:
3460:
3456:
3452:
3442:
3440:
3436:
3432:
3428:
3424:
3410:
3406:
3404:
3400:
3396:
3395:combinatorics
3389:Combinatorics
3386:
3384:
3380:
3376:
3375:Haar measures
3372:
3366:
3356:
3354:
3351:treatment of
3350:
3346:
3342:
3338:
3334:
3330:
3306:
3298:
3295:
3291:
3287:
3284:
3280:
3268:
3264:
3260:
3258:
3249:
3245:
3241:
3234:
3231:
3228:
3224:
3212:
3211:
3210:
3208:
3204:
3199:
3189:
3187:
3183:
3179:
3175:
3171:
3167:
3163:
3158:
3154:
3149:
3136:
3132:
3126:
3123:
3119:
3115:
3111:
3105:
3101:
3099:
3095:
3091:
3087:
3083:
3079:
3075:
3071:
3067:
3063:
3059:
3055:
3050:
3040:
3038:
3034:
3030:
3023:
3019:
3015:
3011:
3007:
3006:Galois theory
3002:
3001:Galois theory
2995:Galois theory
2992:
2990:
2986:
2982:
2972:
2970:
2966:
2962:
2957:
2955:
2951:
2947:
2944:
2940:
2932:
2914:
2909:
2887:
2865:
2862:
2859:
2856:
2851:
2847:
2838:
2835:
2831:
2827:
2823:
2820:
2816:
2812:
2808:
2804:
2800:
2796:
2792:
2789:
2785:
2781:
2780:
2779:
2777:
2773:
2767:
2757:
2755:
2751:
2747:
2743:
2739:
2736:
2732:
2728:
2724:
2720:
2715:
2707:
2703:
2689:
2683:
2677:
2671:
2668:
2663:
2659:
2655:
2652:
2649:
2646:
2640:
2637:
2617:
2614:
2611:
2608:
2600:
2584:
2578:
2575:
2572:
2569:
2566:
2563:
2560:
2557:
2554:
2551:
2548:
2537:
2533:
2529:
2525:
2524:
2518:
2516:
2512:
2507:
2503:
2501:
2485:
2477:
2447:
2444:
2440:
2434:
2431:
2427:
2423:
2420:
2417:
2414:
2411:
2408:
2385:
2379:
2376:
2373:
2362:
2358:
2354:
2351:generated by
2350:
2332:
2329:
2326:
2316:
2312:
2300:
2296:
2292:
2287:
2283:
2278:
2272:
2262:
2260:
2256:
2255:Galois theory
2252:
2248:
2244:
2240:
2236:
2232:
2227:
2225:
2224:Arthur Tresse
2221:
2217:
2213:
2209:
2205:
2201:
2197:
2191:
2181:
2179:
2178:
2173:
2170:
2166:
2162:
2158:
2154:
2150:
2146:
2142:
2138:
2134:
2129:
2127:
2126:Schur's lemma
2123:
2119:
2115:
2111:
2107:
2103:
2099:
2095:
2091:
2086:
2084:
2080:
2074:
2070:
2066:
2062:
2058:
2054:
2049:
2045:
2042:
2038:
2034:
2030:
2026:
2022:
2003:
1997:
1991:
1988:
1982:
1979:
1976:
1969:
1968:
1967:
1965:
1961:
1958:
1954:
1950:
1946:
1942:
1938:
1934:
1933:
1929:
1923:
1913:
1911:
1907:
1903:
1899:
1895:
1891:
1887:
1883:
1882:finite fields
1879:
1875:
1871:
1867:
1862:
1860:
1859:simple groups
1856:
1852:
1848:
1844:
1838:
1823:
1821:
1817:
1815:
1810:
1807:
1805:
1800:
1796:
1792:
1788:
1784:
1780:
1776:
1772:
1768:
1764:
1760:
1755:
1753:
1749:
1745:
1741:
1737:
1733:
1729:
1710:
1705:
1702:
1698:
1691:
1688:
1685:
1679:
1676:
1673:
1669:
1666:
1663:
1654:
1651:
1648:
1642:
1639:
1633:
1630:
1627:
1624:
1621:
1614:
1613:
1612:
1611:(inversion),
1610:
1606:
1602:
1598:
1594:
1590:
1580:
1578:
1574:
1570:
1566:
1562:
1558:
1557:simple groups
1554:
1550:
1549:finite groups
1546:
1541:
1539:
1535:
1531:
1527:
1523:
1520:. If a group
1519:
1518:number theory
1515:
1511:
1507:
1504:
1500:
1497:, of a group
1496:
1492:
1488:
1484:
1465:
1459:
1451:
1445:
1442:
1435:
1434:
1433:
1431:
1427:
1423:
1419:
1408:
1406:
1402:
1398:
1394:
1390:
1386:
1382:
1378:
1373:
1371:
1367:
1363:
1359:
1355:
1351:
1347:
1337:
1335:
1331:
1327:
1323:
1319:
1316:
1312:
1308:
1304:
1300:
1299:linear groups
1296:
1295:matrix groups
1289:Matrix groups
1286:
1284:
1280:
1273:
1269:
1264:
1259:
1253:
1247:
1245:
1240:
1236:
1231:
1227:
1223:
1219:
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1209:
1205:
1201:
1197:
1194:elements and
1193:
1189:
1185:
1181:
1177:
1173:
1169:
1165:
1161:
1157:
1153:
1149:
1139:
1137:
1133:
1129:
1125:
1124:matrix groups
1119:
1109:
1107:
1106:simple groups
1104:
1100:
1096:
1092:
1086:
1084:
1080:
1076:
1072:
1068:
1064:
1060:
1056:
1052:
1048:
1044:
1040:
1039:Arthur Cayley
1036:
1032:
1028:
1026:
1022:
1018:
1014:
1010:
1006:
1005:Galois theory
1002:
998:
994:
990:
986:
982:
978:
974:
970:
966:
962:
961:number theory
956:
946:
944:
939:
934:
932:
928:
924:
920:
916:
912:
908:
904:
903:hydrogen atom
900:
895:
893:
889:
885:
881:
877:
876:vector spaces
873:
869:
865:
861:
857:
853:
845:
841:
837:
833:
828:
817:
812:
810:
805:
803:
798:
797:
795:
794:
787:
784:
783:
780:
777:
776:
773:
770:
769:
766:
763:
762:
759:
754:
753:
743:
740:
737:
736:
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728:
725:
723:
720:
719:
716:
713:
711:
708:
706:
703:
702:
699:
693:
691:
685:
683:
677:
675:
669:
667:
661:
660:
656:
652:
649:
648:
644:
640:
637:
636:
632:
628:
625:
624:
620:
616:
613:
612:
608:
604:
601:
600:
596:
592:
589:
588:
584:
580:
577:
576:
572:
568:
565:
564:
561:
558:
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545:
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539:
532:
529:
527:
524:
522:
519:
518:
490:
465:
464:
462:
456:
453:
428:
425:
424:
418:
415:
413:
410:
409:
405:
404:
393:
390:
388:
385:
382:
379:
378:
377:
376:
373:
370:
369:
364:
361:
360:
357:
354:
353:
350:
347:
345:
343:
339:
338:
335:
332:
330:
327:
326:
323:
320:
318:
315:
314:
313:
312:
306:
303:
300:
295:
292:
291:
287:
282:
279:
276:
271:
268:
265:
260:
257:
256:
255:
254:
249:
248:Finite groups
244:
243:
232:
229:
227:
224:
223:
222:
221:
216:
213:
211:
208:
206:
203:
201:
198:
196:
193:
191:
188:
186:
183:
181:
178:
176:
173:
171:
168:
166:
163:
162:
161:
160:
155:
152:
150:
147:
146:
145:
144:
141:
140:
136:
135:
130:
127:
125:
122:
120:
117:
115:
112:
109:
107:
104:
103:
102:
101:
96:
93:
91:
88:
86:
83:
82:
81:
80:
75:Basic notions
72:
71:
67:
63:
62:
59:
54:
50:
46:
45:
40:
33:
19:
5256:Group theory
5232:
5220:
5208:
5189:
5122:Optimization
4984:Differential
4908:Differential
4875:Order theory
4870:Graph theory
4774:Group theory
4773:
4621:
4600:
4546:
4513:
4495:Group Theory
4494:
4472:
4446:
4430:
4398:
4366:
4339:
4299:
4293:
4279:
4247:
4243:
4232:the original
4219:
4215:
4179:
4175:
4144:
4105:
4073:
4033:
4025:
3993:math/9904135
3983:
3977:
3971:
3955:
3944:
3933:
3915:
3911:
3901:
3888:
3871:
3860:group object
3854:
3844:, retrieved
3840:the original
3833:
3823:
3775:uses finite
3754:
3741:
3739:cyclic group
3729:Cryptography
3722:
3716:
3713:
3701:
3694:
3691:
3685:
3682:
3676:
3673:
3669:
3667:
3659:
3652:
3634:
3617:
3613:
3607:
3604:
3596:
3590:
3587:
3583:
3579:
3573:
3570:
3566:
3560:
3545:
3519:to classify
3517:space groups
3509:point groups
3498:
3480:
3468:gauge theory
3448:
3420:
3392:
3385:techniques.
3368:
3341:class groups
3326:
3201:
3151:
3130:
3124:
3121:
3117:
3113:
3109:
3088:. Similarly
3052:
3021:
3018:Galois group
3004:
2978:
2958:
2948:
2936:
2931:Galois group
2818:
2803:metric space
2794:
2783:
2771:
2769:
2753:
2745:
2737:
2735:metric space
2730:
2719:Cayley graph
2712:
2598:
2523:word problem
2521:
2519:
2504:
2499:
2360:
2356:
2352:
2298:
2295:presentation
2294:
2290:
2285:
2281:
2274:
2228:
2219:
2193:
2176:
2171:
2140:
2132:
2130:
2121:
2109:
2105:
2097:
2093:
2089:
2087:
2082:
2078:
2072:
2068:
2064:
2060:
2056:
2052:
2050:) such that
2047:
2043:
2041:automorphism
2036:
2032:
2024:
2018:
1959:
1957:vector space
1952:
1948:
1944:
1940:
1936:
1930:
1927:
1925:
1863:
1843:local theory
1840:
1837:Finite group
1819:
1813:
1808:
1803:
1794:
1790:
1786:
1778:
1769:and unitary
1756:
1739:
1725:
1608:
1604:
1588:
1586:
1577:Emmy Noether
1542:
1537:
1533:
1529:
1525:
1521:
1510:Class groups
1505:
1498:
1494:
1490:
1483:factor group
1482:
1480:
1429:
1426:presentation
1417:
1414:
1374:
1369:
1362:vector space
1357:
1353:
1349:
1343:
1333:
1325:
1321:
1317:
1310:
1302:
1294:
1292:
1278:
1262:
1251:
1248:
1238:
1234:
1225:
1217:
1212:
1203:
1199:
1198:consists of
1195:
1191:
1190:consists of
1187:
1183:
1175:
1172:permutations
1171:
1167:
1159:
1155:
1145:
1128:presentation
1121:
1087:
1029:
1015:and, later,
1009:field theory
958:
935:
896:
858:studies the
856:group theory
855:
849:
832:Rubik's Cube
830:The popular
654:
642:
630:
618:
606:
594:
582:
570:
341:
298:
285:
274:
263:
259:Cyclic group
137:
124:Free product
95:Group action
58:Group theory
57:
53:Group theory
52:
39:Social group
18:Group Theory
5234:WikiProject
5077:Game theory
5057:Probability
4794:Homological
4784:Multilinear
4764:Commutative
4741:Type theory
4708:Foundations
4664:mathematics
3793:braid group
3709:tetrahedral
3423:periodicity
3399:permutation
3273: prime
3066:deformation
2824:If instead
2723:word metric
2277:group table
2218:. The term
1898:Weyl groups
1545:isomorphism
1283:in radicals
1178:is a group
1063:Felix Klein
1047:geometrical
1021:Felix Klein
544:Topological
383:alternating
5062:Statistics
4941:Arithmetic
4903:Arithmetic
4769:Elementary
4736:Set theory
4384:symmetries
4222:: 239–50,
4095:References
3846:2011-12-20
3791:such as a
3759:serve for
3747:underlies
3707:and other
3474:, and the
3381:and other
3058:associates
2630:, one has
2513:via their
2349:free group
2239:structures
2226:, page 3.
2212:Sophus Lie
2190:Lie theory
2184:Lie theory
2153:characters
2149:Lie groups
1890:Lie groups
1763:Lie groups
1761:, whereas
1728:continuous
1573:Emil Artin
1405:continuous
1164:bijections
1146:The first
1132:generators
1075:Lie groups
1071:Sophus Lie
1055:hyperbolic
936:The early
892:Lie groups
880:operations
836:Ernő Rubik
651:Symplectic
591:Orthogonal
548:Lie groups
455:Free group
180:continuous
119:Direct sum
4989:Geometric
4979:Algebraic
4918:Euclidean
4893:Algebraic
4789:Universal
4361:Livio, M.
4316:0025-570X
4228:0010-437X
3563:chemistry
3529:chirality
3501:chemistry
3327:captures
3296:−
3288:−
3265:∏
3232:≥
3225:∑
2965:morphisms
2910:−
2857:−
2807:bijection
2788:bijective
2687:⟩
2681:⟨
2678:≅
2675:⟩
2656:∣
2644:⟨
2641:≅
2582:⟩
2558:∣
2546:⟨
2532:algorithm
2478:×
2453:⟩
2445:−
2432:−
2418:∣
2406:⟨
2383:⟩
2377:∣
2371:⟨
2330:∈
2196:Lie group
1992:
1986:→
1977:ρ
1935:on a set
1910:chemistry
1870:Steinberg
1866:Chevalley
1748:Lie group
1703:−
1695:↦
1683:→
1661:↦
1637:→
1631:×
1463:⟩
1449:⟨
1389:manifolds
1136:relations
1051:euclidean
923:chemistry
862:known as
715:Conformal
603:Euclidean
210:nilpotent
5250:Category
5210:Category
4966:Topology
4913:Discrete
4898:Analytic
4885:Geometry
4857:Discrete
4812:Calculus
4804:Analysis
4759:Abstract
4698:Glossary
4681:Timeline
4610:(1911),
4573:36131259
4545:(1994),
4471:(2001),
4429:, 2006.
4427:Ronan M.
4397:(1970),
4363:(2005),
4288:license.
4271:groupoid
4204:18226463
4182:: 3–12,
4104:(1991),
4018:18211120
3959:See the
3918:: 1–88,
3894:faithful
3875:Such as
3864:category
3799:See also
3630:hydrogen
3616:, where
3349:Kummer's
3329:the fact
3129:, where
2961:category
2939:symmetry
2811:isometry
2776:symmetry
2077:for any
1886:symmetry
1847:solvable
1750:, or an
1401:discrete
1307:matrices
1222:subgroup
1079:analytic
989:Lagrange
979:work on
969:geometry
901:and the
899:crystals
710:Poincaré
555:Solenoid
427:Integers
417:Lattices
392:sporadic
387:Lie type
215:solvable
205:dihedral
190:additive
175:infinite
85:Subgroup
5222:Commons
5004:Applied
4974:General
4751:Algebra
4676:History
4620:(ed.),
4565:1269324
4536:0347778
4388:physics
4332:0863090
4324:2690312
4266:2223010
4196:0290613
4168:2504193
4136:1102012
4066:2738874
4010:1896232
3705:methane
3451:physics
3445:Physics
3425:in the
2744:, then
1816:-groups
1783:lattice
1736:regular
1569:Hilbert
1313:over a
1301:. Here
1206:is the
993:Ruffini
977:Gauss's
949:History
919:physics
705:Lorentz
627:Unitary
526:Lattice
466:PSL(2,
200:abelian
111:(Semi-)
4923:Finite
4779:Linear
4686:Future
4662:Major
4571:
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4088:, Ch 2
4084:
4064:
4054:
4016:
4008:
3626:oxygen
3601:chiral
3515:, and
3470:, the
3333:primes
3027:, the
2943:closed
2826:angles
2799:metric
2727:Milnor
2511:graphs
2347:, the
2019:where
1732:smooth
1268:simple
1256:, the
1230:Cayley
1180:acting
1103:finite
1031:Galois
995:, and
967:, and
925:, and
905:, and
884:axioms
874:, and
872:fields
864:groups
842:. See
560:Circle
491:SL(2,
380:cyclic
344:-group
195:cyclic
170:finite
165:simple
149:kernel
5150:lists
4693:Lists
4666:areas
4616:, in
4320:JSTOR
4200:S2CID
4062:S2CID
4014:S2CID
3988:arXiv
3816:Notes
3622:water
3417:Music
3178:torus
2985:Rings
2954:graph
2200:group
2198:is a
2124:(via
2116:(see
1962:is a
1955:on a
1880:over
1742:is a
1599:, or
1501:by a
1485:, or
1385:Klein
1360:is a
1315:field
1297:, or
1220:is a
1186:. If
1148:class
868:rings
744:Sp(∞)
741:SU(∞)
154:image
4601:Plus
4569:OCLC
4551:ISBN
4522:ISBN
4499:ISBN
4481:ISBN
4451:ISBN
4435:ISBN
4417:OCLC
4407:ISBN
4373:ISBN
4348:ISBN
4312:ISSN
4286:GFDL
4224:ISSN
4154:ISBN
4122:ISBN
4082:ISBN
4052:ISBN
3737:The
3527:and
3503:and
3343:and
3172:are
2902:and
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2059:) ∘
1932:acts
1908:and
1868:and
1849:and
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890:and
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727:Loop
546:and
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1628:G
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1622:m
1609:i
1605:m
1589:G
1538:X
1534:H
1532:/
1530:G
1526:X
1522:G
1506:H
1499:G
1495:H
1493:/
1491:G
1466:.
1460:R
1456:|
1452:S
1446:=
1443:G
1358:X
1354:X
1350:X
1334:G
1326:K
1322:n
1318:K
1311:n
1303:G
1279:n
1263:n
1260:A
1252:n
1239:G
1235:X
1226:X
1218:G
1213:n
1210:S
1204:G
1196:G
1192:n
1188:X
1184:X
1176:G
1168:X
1160:G
1156:X
846:.
815:e
808:t
801:v
697:8
695:E
689:7
687:E
681:6
679:E
673:4
671:F
665:2
663:G
657:)
655:n
645:)
643:n
633:)
631:n
621:)
619:n
609:)
607:n
597:)
595:n
585:)
583:n
573:)
571:n
513:)
500:Z
488:)
475:Z
451:)
438:Z
429:(
342:p
307:Q
299:n
296:D
286:n
283:A
275:n
272:S
264:n
261:Z
41:.
34:.
20:)
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