2171:
The following table lists some examples of presentations for commonly studied groups. Note that in each case there are many other presentations that are possible. The presentation listed is not necessarily the most efficient one possible.
4908:
3577:
3153:
3474:
3050:
4379:
2610:
4618:
824:
957:
397:
1062:
4010:
3913:
3816:
3719:
2390:
4107:
4711:
4207:
2791:
1874:. In this perspective, we declare two words to be equivalent if it is possible to get from one to the other by a sequence of moves, where each move consists of adding or removing a consecutive pair
2481:
2119:
states that a finitely generated group has a recursive presentation if and only if it can be embedded in a finitely presented group. From this we can deduce that there are (up to isomorphism) only
3226:
4469:
2228:
3387:
2925:
1417:
223:
2687:
4956:
271:
2289:
2001:) if it has a presentation that is finitely generated (respectively finitely related, a finite presentation). A group which has a finite presentation with a single relation is called a
118:
2838:
1540:
1504:
1460:
1355:
1254:
1774:
3619:
3267:
2966:
1824:
5646:
1872:
1620:
1938:
1905:
1665:
1089:
used in this article for a presentation is now the most common, earlier writers used different variations on the same format. Such notations include the following:
5684:
1711:
1323:
1281:
1222:
4726:
1569:
3481:
3057:
283:
Every group has a presentation, and in fact many different presentations; a presentation is often the most compact way of describing the structure of the group.
2115:
Every finitely presented group is recursively presented, but there are recursively presented groups that cannot be finitely presented. However a theorem of
3397:
2973:
4240:
1948:, or by adding or removing a consecutive copy of a relator. The group elements are the equivalence classes, and the group operation is concatenation.
5595:
2503:
5869:
5625:
4508:
276:
thanks to the convention that terms that do not include an equals sign are taken to be equal to the group identity. Such terms are called
728:
5921:
873:
5876:― fundamental algorithms from theoretical computer science, computational number theory, and computational commutative algebra, etc.
5838:
5815:
5771:
316:
984:
3935:
3838:
3741:
3644:
2315:
4030:
287:
4642:
5926:
4131:
2720:
2131:
many non-isomorphic two generator groups. Therefore, there are finitely generated groups that cannot be recursively presented.
5799:
5846:
5737:
2419:
5672:
5154:
3164:
1288:
156:
4416:
5853:
2201:
3340:
2878:
1366:
4635:
5822:― This useful reference has tables of presentations of all small finite groups, the reflection groups, and so forth.
180:
5763:
2642:
2160:
1952:
4931:
234:
2255:
66:
70:—so that every element of the group can be written as a product of powers of some of these generators—and a set
5803:
5457:
5452:
5249:
4922:
1965:
1156:
37:
This article is about specifying generators and relations of a group. For describing a module over a ring, see
2139:
One of the earliest presentations of a group by generators and relations was given by the Irish mathematician
88:
2811:
1513:
1477:
1433:
1328:
1227:
5118:
2412:
5586:
5578:
5574:
5447:
5398:
5380:
2140:
2077:
1724:
5467:
5888:
3586:
3234:
2933:
962:
An even shorter form drops the equality and identity signs, to list just the set of relators, which is
1779:
5693:
5532:
5462:
5406:
5064:
3928:
561:
Informally, we can consider these products on the left hand side as being elements of the free group
31:
5826:
5352:
1186:
53:
1841:
1586:
5556:
4998:
4994:
3583:
Note the similarity with the symmetric group; the only difference is the removal of the relation
2800:
5520:
2026:
2104:. This usage may seem odd, but it is possible to prove that if a group has a presentation with
5865:
5834:
5811:
5767:
5721:
5621:
5617:
5548:
3734:
2148:
1835:
140:
38:
4903:{\displaystyle \langle a,b\mid a^{2},b^{3},(ab)^{13},^{5},^{4},((ab)^{4}ab^{-1})^{6}\rangle }
1910:
1877:
1637:
5777:
5729:
5711:
5701:
5655:
5540:
5501:
5492:
Peifer, David (1997). "An
Introduction to Combinatorial Group Theory and the Word Problem".
5435:
5023:
4501:
4023:
3831:
2615:
2152:
2144:
2124:
3572:{\displaystyle \sigma _{i}\sigma _{i+1}\sigma _{i}=\sigma _{i+1}\sigma _{i}\sigma _{i+1}\ }
3148:{\displaystyle \sigma _{i}\sigma _{i+1}\sigma _{i}=\sigma _{i+1}\sigma _{i}\sigma _{i+1}\ }
1689:
1301:
1259:
1200:
5781:
5733:
5659:
4392:
2865:
5845:― Schreier's method, Nielsen's method, free presentations, subgroups and HNN extensions,
1548:
526:. Each such product equivalence can be expressed as an equality to the identity, such as
5697:
5536:
5525:
Proceedings of the Royal
Society of London. Series A. Mathematical and Physical Sciences
5892:
5858:
4926:
2496:
2304:
1358:
706:
462:
144:
5716:
3469:{\displaystyle \sigma _{i}\sigma _{j}=\sigma _{j}\sigma _{i}{\mbox{ if }}j\neq i\pm 1}
3045:{\displaystyle \sigma _{i}\sigma _{j}=\sigma _{j}\sigma _{i}{\mbox{ if }}j\neq i\pm 1}
5915:
5755:
5641:
5560:
5428:
5424:
5150:
4409:
3637:
2116:
2073:
2025:
or a finite subset of them, then it is easy to set up a simple one to one coding (or
431:, subject only to canceling a generator with an adjacent occurrence of its inverse.
5505:
5419:
5402:
5386:
5211:
5106:
4478:
4374:{\displaystyle \langle a,b,j\mid aba=bab,(aba)^{4},j^{2},(ja)^{2},(jb)^{2}\rangle }
2244:
1679:
167:
427:
may be zero). In less formal terms, the group consists of words in the generators
17:
5676:
5644:(1955), "On the algorithmic unsolvability of the word problem in group theory",
5410:
5392:
3332:
2156:
2128:
45:
5754:
Johnson, D.L.; Robertson, E.L. (1979). "Finite groups of deficiency zero". In
5291:
4913:
4719:
4220:
2845:
2190:
1174:
299:
148:
5552:
1834:
The definition of group presentation may alternatively be recast in terms of
5897:
5706:
2120:
5725:
5544:
2605:{\displaystyle \langle r,f\mid r^{2n},r^{n}=f^{2},frf^{-1}=r^{-1}\rangle }
4959:
280:, distinguishing them from the relations that do include an equals sign.
2232:
A free group is "free" in the sense that it is subject to no relations.
617:, each of which is also equivalent to 1 when considered as products in D
4613:{\displaystyle \langle x,y,z\mid z=xyx^{-1}y^{-1},xz=zx,yz=zy\rangle }
2050:
to the natural numbers, such that we can find algorithms that, given
228:
where 1 is the group identity. This may be written equivalently as
127:
has the above presentation if it is the "freest group" generated by
1630:
are supposed to be equal in the quotient group. Thus, for example,
819:{\displaystyle \langle r,f\mid r^{8}=1,f^{2}=1,(rf)^{2}=1\rangle .}
450:
is also of the above form; but in general, these products will not
4224:
5762:. London Mathematical Society Lecture Note Series. Vol. 36.
5397:
A presentation of a group determines a geometry, in the sense of
27:
Specification of a mathematical group by generators and relations
952:{\displaystyle \langle r,f\mid r^{8}=f^{2}=(rf)^{2}=1\rangle .}
685:
of members of such conjugates. It follows that each element of
5438:) are intrinsic, meaning independent of choice of generators.
5347:), is the maximum of the deficiency over all presentations of
5105:
One may take the elements of the group for generators and the
1283:
by the smallest normal subgroup that contains each element of
392:{\displaystyle s_{1}^{a_{1}}s_{2}^{a_{2}}\cdots s_{n}^{a_{n}}}
1057:{\displaystyle \langle r,f\mid r^{8},f^{2},(rf)^{2}\rangle .}
4005:{\displaystyle \langle s,t\mid s^{2},t^{3},(st)^{5}\rangle }
3908:{\displaystyle \langle s,t\mid s^{2},t^{3},(st)^{4}\rangle }
3811:{\displaystyle \langle s,t\mid s^{2},t^{3},(st)^{3}\rangle }
3714:{\displaystyle \langle s,t\mid s^{2},t^{2},(st)^{2}\rangle }
2385:{\displaystyle \langle r,f\mid r^{n},f^{2},(rf)^{2}\rangle }
5149:
describe the same element in the group. This was shown by
4102:{\displaystyle \langle i,j\mid i^{4},jij=i,iji=j\rangle \,}
5351:. The deficiency of a finite group is non-positive. The
4706:{\displaystyle \langle a,b\mid a^{n}=ba^{m}b^{-1}\rangle }
1951:
This point of view is particularly common in the field of
4202:{\displaystyle \langle a,b\mid aba=bab,(aba)^{4}\rangle }
2786:{\displaystyle \langle x,y\mid x^{m},y^{n},xy=yx\rangle }
2088:) is recursive (respectively recursively enumerable). If
5906:
5864:(1st ed.). Cambridge: Cambridge University Press.
5833:(2nd ed.). Cambridge: Cambridge University Press.
5133:
for which there is no algorithm which, given two words
5579:"Memorandum respecting a new System of Roots of Unity"
5086:. This presentation may be highly inefficient if both
3445:
3021:
2476:{\displaystyle \langle r,f\mid f^{2},(rf)^{2}\rangle }
2123:
many finitely generated recursively presented groups.
1545:
It is a common practice to write relators in the form
643:, then it follows by definition that every element of
286:
A closely related but different concept is that of an
4934:
4729:
4645:
4511:
4419:
4243:
4134:
4033:
3938:
3841:
3744:
3647:
3589:
3484:
3400:
3343:
3237:
3167:
3060:
2976:
2936:
2881:
2814:
2723:
2645:
2506:
2422:
2318:
2258:
2204:
1913:
1880:
1844:
1782:
1727:
1692:
1640:
1589:
1551:
1516:
1480:
1436:
1369:
1331:
1304:
1262:
1230:
1203:
987:
876:
731:
319:
237:
183:
91:
3221:{\displaystyle {(\sigma _{i}\sigma _{i+1}})^{3}=1\ }
2108:
recursively enumerable then it has another one with
1622:. This has the intuitive meaning that the images of
5647:
Proceedings of the
Steklov Institute of Mathematics
4464:{\displaystyle \langle a,b\mid a^{2},b^{3}\rangle }
3158:The last set of relations can be transformed into
2174:
2096:recursively enumerable, then the presentation is a
5907:Small groups and their presentations on GroupNames
5857:
4950:
4902:
4705:
4612:
4463:
4373:
4201:
4101:
4004:
3907:
3810:
3713:
3613:
3571:
3468:
3381:
3261:
3220:
3147:
3044:
2960:
2919:
2832:
2785:
2681:
2604:
2475:
2384:
2283:
2223:{\displaystyle \langle S\mid \varnothing \rangle }
2222:
1932:
1899:
1866:
1818:
1768:
1705:
1659:
1614:
1563:
1534:
1498:
1454:
1411:
1349:
1317:
1275:
1248:
1216:
1056:
951:
818:
391:
310:described as a finite length product of the form:
265:
217:
112:
3382:{\displaystyle \sigma _{1},\ldots ,\sigma _{n-1}}
2920:{\displaystyle \sigma _{1},\ldots ,\sigma _{n-1}}
2159:, in the early 1880s, laying the foundations for
1981:is finite. If both are finite it is said to be a
469:of order sixteen can be generated by a rotation,
1412:{\displaystyle \langle S\mid R\rangle =F_{S}/N.}
139:is said to have the above presentation if it is
5685:Proceedings of the National Academy of Sciences
5037:, therefore is equal to its normal closure, so
5153:in 1955 and a different proof was obtained by
1155:"Relator" redirects here. For other uses, see
218:{\displaystyle \langle a\mid a^{n}=1\rangle ,}
5101:Every finite group has a finite presentation.
2682:{\displaystyle \langle x,y\mid xy=yx\rangle }
8:
5808:Generators and Relations for Discrete Groups
5434:Further, some properties of this graph (the
5121:states that there is a finite presentation
4951:{\displaystyle \mathbf {Z} \wr \mathbf {Z} }
4897:
4730:
4700:
4646:
4607:
4512:
4458:
4420:
4368:
4244:
4196:
4135:
4095:
4034:
3999:
3939:
3902:
3842:
3805:
3745:
3708:
3648:
2827:
2815:
2780:
2724:
2676:
2646:
2599:
2507:
2470:
2423:
2379:
2319:
2278:
2259:
2217:
2205:
2062:, and vice versa. We can then call a subset
1674:, it is possible to build a presentation of
1529:
1517:
1493:
1481:
1449:
1437:
1382:
1370:
1344:
1332:
1243:
1231:
1048:
988:
943:
877:
810:
732:
477:, of order 2; and certainly any element of D
266:{\displaystyle \langle a\mid a^{n}\rangle ,}
257:
238:
209:
184:
104:
92:
5413:. These are also two resulting orders, the
2284:{\displaystyle \langle a\mid a^{n}\rangle }
1634:in the list of relators is equivalent with
5860:Computation with Finitely Presented Groups
2151:. The first systematic study was given by
5715:
5705:
4943:
4935:
4933:
4891:
4878:
4865:
4840:
4809:
4784:
4762:
4749:
4728:
4691:
4681:
4665:
4644:
4562:
4549:
4510:
4452:
4439:
4418:
4362:
4340:
4318:
4305:
4242:
4190:
4133:
4098:
4053:
4032:
3993:
3971:
3958:
3937:
3896:
3874:
3861:
3840:
3799:
3777:
3764:
3743:
3702:
3680:
3667:
3646:
3599:
3594:
3588:
3554:
3544:
3528:
3515:
3499:
3489:
3483:
3444:
3438:
3428:
3415:
3405:
3399:
3367:
3348:
3342:
3247:
3242:
3236:
3203:
3186:
3176:
3168:
3166:
3130:
3120:
3104:
3091:
3075:
3065:
3059:
3020:
3014:
3004:
2991:
2981:
2975:
2946:
2941:
2935:
2905:
2886:
2880:
2813:
2756:
2743:
2722:
2644:
2590:
2574:
2555:
2542:
2526:
2505:
2464:
2442:
2421:
2373:
2351:
2338:
2317:
2272:
2257:
2203:
1918:
1912:
1888:
1879:
1855:
1843:
1826:is an entry in the multiplication table.
1810:
1797:
1787:
1781:
1757:
1752:
1742:
1732:
1726:
1697:
1691:
1645:
1639:
1594:
1588:
1550:
1515:
1479:
1435:
1398:
1392:
1368:
1330:
1309:
1303:
1267:
1261:
1229:
1208:
1202:
1042:
1020:
1007:
986:
931:
909:
896:
875:
798:
770:
751:
730:
693:, will also evaluate to 1; and thus that
602:be the subgroup generated by the strings
381:
376:
371:
356:
351:
346:
334:
329:
324:
318:
251:
236:
197:
182:
90:
5521:"Subgroups of finitely presented groups"
5487:
5485:
5483:
5059:. Since the identity map is surjective,
1067:All three presentations are equivalent.
113:{\displaystyle \langle S\mid R\rangle .}
5479:
5282:, where means that every element from
4111:For an alternative presentation see Dic
2833:{\displaystyle \langle S\mid R\rangle }
2214:
1535:{\displaystyle \langle S\mid R\rangle }
1499:{\displaystyle \langle S\mid R\rangle }
1455:{\displaystyle \langle S\mid R\rangle }
1350:{\displaystyle \langle S\mid R\rangle }
1249:{\displaystyle \langle S\mid R\rangle }
4997:of free groups, there exists a unique
4925:that is not finitely presented is the
2021:consisting of all the natural numbers
978:. Doing this gives the presentation
867:abbreviated, giving the presentation
306:is a group where each element can be
7:
1769:{\displaystyle g_{i}g_{j}g_{k}^{-1}}
1224:. To form a group with presentation
78:among those generators. We then say
689:, when considered as a product in D
3282:is the permutation that swaps the
25:
5427:. An important example is in the
5286:commutes with every element from
3614:{\displaystyle \sigma _{i}^{2}=1}
3262:{\displaystyle \sigma _{i}^{2}=1}
2961:{\displaystyle \sigma _{i}^{2}=1}
1474:is said to have the presentation
5601:from the original on 2003-06-26.
5094:are much larger than necessary.
4944:
4936:
1819:{\displaystyle g_{i}g_{j}=g_{k}}
1287:. (This subgroup is called the
288:absolute presentation of a group
5743:from the original on 2015-09-24
4974:Every group has a presentation.
2100:and the corresponding group is
844:}, and the set of relations is
498:However, we have, for example,
5506:10.1080/0025570X.1997.11996491
5339:of a finitely presented group
5063:is also surjective, so by the
4888:
4862:
4852:
4849:
4837:
4818:
4806:
4793:
4781:
4771:
4359:
4349:
4337:
4327:
4302:
4289:
4187:
4174:
3990:
3980:
3893:
3883:
3796:
3786:
3699:
3689:
3200:
3169:
2461:
2451:
2370:
2360:
1039:
1029:
928:
918:
829:Here the set of generators is
795:
785:
131:subject only to the relations
52:is one method of specifying a
1:
5810:. New York: Springer-Verlag.
5117:The negative solution to the
1963:A presentation is said to be
518:, etc., so such products are
5371:if this number is required.
2009:Recursively presented groups
1867:{\displaystyle S\cup S^{-1}}
1721:to be all words of the form
1615:{\displaystyle y^{-1}x\in R}
1197:naturally gives a subset of
632:generated by all conjugates
409:are elements of S, adjacent
157:normal subgroup generated by
56:. A presentation of a group
5614:Mathematics and its history
5026:of this homomorphism. Then
4978:To see this, given a group
423:are non-zero integers (but
5943:
5922:Combinatorial group theory
5764:Cambridge University Press
5575:Sir William Rowan Hamilton
5390:
5384:
5378:
4982:, consider the free group
2398:represents a rotation and
2161:combinatorial group theory
1953:combinatorial group theory
1680:group multiplication table
1583:. What this means is that
1154:
473:, of order 8; and a flip,
442:is a generating subset of
36:
29:
5847:Golod–Shafarevich theorem
5519:Higman, G. (1961-08-08).
5359:can be generated by −def(
5307:of a finite presentation
5065:First Isomorphism Theorem
5018:is the identity map. Let
3308:is a 3-cycle on the set {
2127:has shown that there are
1959:Finitely presented groups
1838:of words on the alphabet
166:As a simple example, the
5760:Homological Group Theory
5612:Stillwell, John (2002).
5458:Presentation of a monoid
5453:Presentation of a module
4923:finitely generated group
2147:– a presentation of the
2092:is indexed as above and
1157:Relator (disambiguation)
697:is a normal subgroup of
446:, then every element of
30:Not to be confused with
5831:Presentations of Groups
5707:10.1073/pnas.44.10.1061
5119:word problem for groups
4636:Baumslag–Solitar groups
2622:is a special case when
2413:infinite dihedral group
2046:from the free group on
1933:{\displaystyle x^{-1}x}
1900:{\displaystyle xx^{-1}}
1686:to be the set elements
1660:{\displaystyle r^{n}=1}
1357:is then defined as the
1256:, take the quotient of
454:describe an element of
5927:Combinatorics on words
5587:Philosophical Magazine
5545:10.1098/rspa.1961.0132
5448:Nielsen transformation
5399:geometric group theory
5381:Geometric group theory
5375:Geometric group theory
5103:
4976:
4952:
4904:
4707:
4614:
4465:
4375:
4203:
4103:
4006:
3909:
3812:
3715:
3615:
3573:
3470:
3383:
3290:+1st one. The product
3263:
3222:
3149:
3046:
2962:
2921:
2834:
2787:
2683:
2606:
2477:
2386:
2285:
2224:
2141:William Rowan Hamilton
2098:recursive presentation
2078:recursively enumerable
1934:
1901:
1868:
1820:
1770:
1707:
1661:
1616:
1565:
1536:
1500:
1456:
1413:
1351:
1319:
1277:
1250:
1218:
1075:Although the notation
1058:
953:
820:
393:
267:
219:
135:. Formally, the group
114:
5468:Tietze transformation
5391:Further information:
5385:Further information:
5205:being disjoint. Then
5113:Novikov–Boone theorem
5096:
5014:whose restriction to
4969:
4953:
4905:
4708:
4615:
4481:of the cyclic groups
4466:
4376:
4219:can be visualized as
4204:
4104:
4007:
3910:
3813:
3716:
3616:
3574:
3471:
3384:
3264:
3223:
3150:
3047:
2963:
2922:
2835:
2788:
2684:
2607:
2478:
2387:
2286:
2225:
2102:recursively presented
1935:
1902:
1869:
1821:
1771:
1708:
1706:{\displaystyle g_{i}}
1662:
1617:
1566:
1537:
1501:
1457:
1414:
1352:
1320:
1318:{\displaystyle F_{S}}
1278:
1276:{\displaystyle F_{S}}
1251:
1219:
1217:{\displaystyle F_{S}}
1059:
954:
821:
705:is isomorphic to the
394:
268:
220:
174:has the presentation
115:
5893:"Group Presentation"
5766:. pp. 275–289.
5616:. Springer. p.
5494:Mathematics Magazine
5463:Set-builder notation
5423:, and corresponding
4932:
4727:
4643:
4509:
4417:
4241:
4132:
4031:
3936:
3839:
3742:
3645:
3587:
3482:
3398:
3341:
3286:th element with the
3235:
3165:
3058:
2974:
2934:
2879:
2812:
2721:
2643:
2504:
2420:
2316:
2256:
2202:
2017:is indexed by a set
1911:
1878:
1842:
1830:Alternate definition
1780:
1725:
1690:
1638:
1587:
1549:
1514:
1478:
1462:and the elements of
1434:
1367:
1329:
1302:
1260:
1228:
1201:
985:
874:
729:
718:. We then say that D
647:is a finite product
317:
235:
181:
89:
32:Group representation
5698:1958PNAS...44.1061B
5537:1961RSPSA.262..455H
5353:Schur multiplicator
3604:
3252:
2951:
1983:finite presentation
1836:equivalence classes
1765:
1682:, as follows. Take
1670:For a finite group
1564:{\displaystyle x=y}
628:be the subgroup of
598:. That is, we let
388:
363:
341:
5889:de Cornulier, Yves
5677:"The word problem"
5363:) generators, and
5355:of a finite group
5141:, decides whether
4999:group homomorphism
4995:universal property
4948:
4900:
4703:
4610:
4461:
4371:
4199:
4099:
4002:
3905:
3808:
3711:
3611:
3590:
3569:
3466:
3449:
3379:
3259:
3238:
3218:
3145:
3042:
3025:
2958:
2937:
2917:
2844:is the set of all
2830:
2801:free abelian group
2783:
2679:
2602:
2473:
2382:
2281:
2220:
1997:finitely presented
1987:finitely generated
1966:finitely generated
1930:
1897:
1864:
1816:
1766:
1748:
1703:
1657:
1612:
1561:
1532:
1496:
1452:
1409:
1347:
1315:
1273:
1246:
1214:
1054:
949:
816:
438:is any group, and
429:and their inverses
416:are distinct, and
389:
367:
342:
320:
263:
215:
110:
18:Group presentation
5871:978-0-521-13507-8
5800:Coxeter, H. S. M.
5692:(10): 1061–1065,
5673:Boone, William W.
5642:Novikov, Pyotr S.
5627:978-0-387-95336-6
5531:(1311): 455–475.
5262:has presentation
5224:has presentation
5185:has presentation
5169:has presentation
4919:
4918:
3929:icosahedral group
3735:tetrahedral group
3568:
3448:
3217:
3144:
3024:
2149:icosahedral group
2003:one-relator group
1510:is isomorphic to
1166:be a set and let
722:has presentation
461:For example, the
82:has presentation
39:free presentation
16:(Redirected from
5934:
5903:
5902:
5875:
5863:
5854:Sims, Charles C.
5844:
5821:
5786:
5785:
5751:
5745:
5744:
5742:
5719:
5709:
5681:
5669:
5663:
5662:
5638:
5632:
5631:
5609:
5603:
5602:
5600:
5583:
5571:
5565:
5564:
5516:
5510:
5509:
5489:
5334:
5332:
5326:
5318:
5281:
5261:
5243:
5223:
5196:
5180:
5132:
5085:
5058:
5013:
4958:of the group of
4957:
4955:
4954:
4949:
4947:
4939:
4921:An example of a
4909:
4907:
4906:
4901:
4896:
4895:
4886:
4885:
4870:
4869:
4845:
4844:
4814:
4813:
4789:
4788:
4767:
4766:
4754:
4753:
4712:
4710:
4709:
4704:
4699:
4698:
4686:
4685:
4670:
4669:
4619:
4617:
4616:
4611:
4570:
4569:
4557:
4556:
4502:Heisenberg group
4470:
4468:
4467:
4462:
4457:
4456:
4444:
4443:
4380:
4378:
4377:
4372:
4367:
4366:
4345:
4344:
4323:
4322:
4310:
4309:
4208:
4206:
4205:
4200:
4195:
4194:
4117:above with n=2.
4108:
4106:
4105:
4100:
4058:
4057:
4024:quaternion group
4011:
4009:
4008:
4003:
3998:
3997:
3976:
3975:
3963:
3962:
3926:
3914:
3912:
3911:
3906:
3901:
3900:
3879:
3878:
3866:
3865:
3832:octahedral group
3829:
3817:
3815:
3814:
3809:
3804:
3803:
3782:
3781:
3769:
3768:
3732:
3720:
3718:
3717:
3712:
3707:
3706:
3685:
3684:
3672:
3671:
3635:
3620:
3618:
3617:
3612:
3603:
3598:
3578:
3576:
3575:
3570:
3566:
3565:
3564:
3549:
3548:
3539:
3538:
3520:
3519:
3510:
3509:
3494:
3493:
3475:
3473:
3472:
3467:
3450:
3446:
3443:
3442:
3433:
3432:
3420:
3419:
3410:
3409:
3388:
3386:
3385:
3380:
3378:
3377:
3353:
3352:
3268:
3266:
3265:
3260:
3251:
3246:
3227:
3225:
3224:
3219:
3215:
3208:
3207:
3198:
3197:
3196:
3181:
3180:
3154:
3152:
3151:
3146:
3142:
3141:
3140:
3125:
3124:
3115:
3114:
3096:
3095:
3086:
3085:
3070:
3069:
3051:
3049:
3048:
3043:
3026:
3022:
3019:
3018:
3009:
3008:
2996:
2995:
2986:
2985:
2967:
2965:
2964:
2959:
2950:
2945:
2926:
2924:
2923:
2918:
2916:
2915:
2891:
2890:
2839:
2837:
2836:
2831:
2792:
2790:
2789:
2784:
2761:
2760:
2748:
2747:
2688:
2686:
2685:
2680:
2616:quaternion group
2611:
2609:
2608:
2603:
2598:
2597:
2582:
2581:
2560:
2559:
2547:
2546:
2534:
2533:
2482:
2480:
2479:
2474:
2469:
2468:
2447:
2446:
2391:
2389:
2388:
2383:
2378:
2377:
2356:
2355:
2343:
2342:
2290:
2288:
2287:
2282:
2277:
2276:
2229:
2227:
2226:
2221:
2175:
2153:Walther von Dyck
2145:icosian calculus
2143:in 1856, in his
2125:Bernhard Neumann
2045:
1999:
1998:
1991:finitely related
1975:finitely related
1947:
1943:
1939:
1937:
1936:
1931:
1926:
1925:
1906:
1904:
1903:
1898:
1896:
1895:
1873:
1871:
1870:
1865:
1863:
1862:
1825:
1823:
1822:
1817:
1815:
1814:
1802:
1801:
1792:
1791:
1775:
1773:
1772:
1767:
1764:
1756:
1747:
1746:
1737:
1736:
1712:
1710:
1709:
1704:
1702:
1701:
1666:
1664:
1663:
1658:
1650:
1649:
1621:
1619:
1618:
1613:
1602:
1601:
1570:
1568:
1567:
1562:
1541:
1539:
1538:
1533:
1505:
1503:
1502:
1497:
1461:
1459:
1458:
1453:
1422:The elements of
1418:
1416:
1415:
1410:
1402:
1397:
1396:
1356:
1354:
1353:
1348:
1324:
1322:
1321:
1316:
1314:
1313:
1282:
1280:
1279:
1274:
1272:
1271:
1255:
1253:
1252:
1247:
1223:
1221:
1220:
1215:
1213:
1212:
1146:
1145:
1130:
1118:
1105:
1104:
1088:
1087:
1063:
1061:
1060:
1055:
1047:
1046:
1025:
1024:
1012:
1011:
977:
958:
956:
955:
950:
936:
935:
914:
913:
901:
900:
863:. We often see
862:
843:
825:
823:
822:
817:
803:
802:
775:
774:
756:
755:
717:
616:
597:
595:
575:
556:
554:
544:
535:
517:
507:
494:
487:
481:is a product of
398:
396:
395:
390:
387:
386:
385:
375:
362:
361:
360:
350:
340:
339:
338:
328:
272:
270:
269:
264:
256:
255:
224:
222:
221:
216:
202:
201:
119:
117:
116:
111:
60:comprises a set
21:
5942:
5941:
5937:
5936:
5935:
5933:
5932:
5931:
5912:
5911:
5887:
5886:
5883:
5872:
5852:
5841:
5825:
5818:
5804:Moser, W. O. J.
5798:
5795:
5790:
5789:
5774:
5753:
5752:
5748:
5740:
5679:
5671:
5670:
5666:
5640:
5639:
5635:
5628:
5611:
5610:
5606:
5598:
5581:
5573:
5572:
5568:
5518:
5517:
5513:
5491:
5490:
5481:
5476:
5444:
5436:coarse geometry
5395:
5389:
5383:
5377:
5328:
5327:| − |
5322:
5320:
5308:
5301:
5263:
5253:
5225:
5215:
5186:
5170:
5163:
5122:
5115:
5109:for relations.
5068:
5052:
5038:
5035:
5007:
5001:
4987:
4968:
4930:
4929:
4887:
4874:
4861:
4836:
4805:
4780:
4758:
4745:
4725:
4724:
4687:
4677:
4661:
4641:
4640:
4558:
4545:
4507:
4506:
4448:
4435:
4415:
4414:
4393:group extension
4358:
4336:
4314:
4301:
4239:
4238:
4186:
4130:
4129:
4116:
4049:
4029:
4028:
4021:
3989:
3967:
3954:
3934:
3933:
3925:
3921:
3892:
3870:
3857:
3837:
3836:
3828:
3824:
3795:
3773:
3760:
3740:
3739:
3731:
3727:
3698:
3676:
3663:
3643:
3642:
3634:
3630:
3626:
3585:
3584:
3550:
3540:
3524:
3511:
3495:
3485:
3480:
3479:
3434:
3424:
3411:
3401:
3396:
3395:
3389:
3363:
3344:
3339:
3338:
3330:
3307:
3298:
3281:
3233:
3232:
3199:
3182:
3172:
3163:
3162:
3126:
3116:
3100:
3087:
3071:
3061:
3056:
3055:
3010:
3000:
2987:
2977:
2972:
2971:
2932:
2931:
2927:
2901:
2882:
2877:
2876:
2866:symmetric group
2863:
2848:of elements of
2810:
2809:
2752:
2739:
2719:
2718:
2641:
2640:
2621:
2586:
2570:
2551:
2538:
2522:
2502:
2501:
2494:
2460:
2438:
2418:
2417:
2410:
2369:
2347:
2334:
2314:
2313:
2302:
2268:
2254:
2253:
2242:
2200:
2199:
2169:
2137:
2071:
2039:
2030:
2027:Gödel numbering
2011:
1996:
1995:
1961:
1945:
1941:
1914:
1909:
1908:
1884:
1876:
1875:
1851:
1840:
1839:
1832:
1806:
1793:
1783:
1778:
1777:
1738:
1728:
1723:
1722:
1693:
1688:
1687:
1641:
1636:
1635:
1590:
1585:
1584:
1547:
1546:
1512:
1511:
1476:
1475:
1466:are called the
1432:
1431:
1426:are called the
1388:
1365:
1364:
1327:
1326:
1305:
1300:
1299:
1263:
1258:
1257:
1226:
1225:
1204:
1199:
1198:
1171:
1160:
1153:
1135:
1134:
1121:
1108:
1094:
1093:
1077:
1076:
1073:
1038:
1016:
1003:
983:
982:
963:
927:
905:
892:
872:
871:
845:
830:
794:
766:
747:
727:
726:
721:
709:
704:
692:
683:
677:
671:
665:
659:
653:
624:If we then let
620:
614:
593:
577:
562:
552:
548:
539:
530:
525:
509:
499:
492:
485:
480:
468:
421:
414:
407:
377:
352:
330:
315:
314:
296:
247:
233:
232:
193:
179:
178:
87:
86:
42:
35:
28:
23:
22:
15:
12:
11:
5:
5940:
5938:
5930:
5929:
5924:
5914:
5913:
5910:
5909:
5904:
5882:
5881:External links
5879:
5878:
5877:
5870:
5850:
5839:
5827:Johnson, D. L.
5823:
5816:
5794:
5791:
5788:
5787:
5772:
5746:
5664:
5650:(in Russian),
5633:
5626:
5604:
5566:
5511:
5478:
5477:
5475:
5472:
5471:
5470:
5465:
5460:
5455:
5450:
5443:
5440:
5429:Coxeter groups
5425:Hasse diagrams
5405:, which has a
5401:: one has the
5379:Main article:
5376:
5373:
5343:, denoted def(
5300:
5297:
5296:
5295:
5250:direct product
5245:
5162:
5159:
5114:
5111:
5050:
5033:
5005:
4985:
4967:
4964:
4946:
4942:
4938:
4927:wreath product
4917:
4916:
4910:
4899:
4894:
4890:
4884:
4881:
4877:
4873:
4868:
4864:
4860:
4857:
4854:
4851:
4848:
4843:
4839:
4835:
4832:
4829:
4826:
4823:
4820:
4817:
4812:
4808:
4804:
4801:
4798:
4795:
4792:
4787:
4783:
4779:
4776:
4773:
4770:
4765:
4761:
4757:
4752:
4748:
4744:
4741:
4738:
4735:
4732:
4722:
4716:
4715:
4713:
4702:
4697:
4694:
4690:
4684:
4680:
4676:
4673:
4668:
4664:
4660:
4657:
4654:
4651:
4648:
4638:
4623:
4622:
4620:
4609:
4606:
4603:
4600:
4597:
4594:
4591:
4588:
4585:
4582:
4579:
4576:
4573:
4568:
4565:
4561:
4555:
4552:
4548:
4544:
4541:
4538:
4535:
4532:
4529:
4526:
4523:
4520:
4517:
4514:
4504:
4498:
4497:
4471:
4460:
4455:
4451:
4447:
4442:
4438:
4434:
4431:
4428:
4425:
4422:
4412:
4401:
4400:
4381:
4370:
4365:
4361:
4357:
4354:
4351:
4348:
4343:
4339:
4335:
4332:
4329:
4326:
4321:
4317:
4313:
4308:
4304:
4300:
4297:
4294:
4291:
4288:
4285:
4282:
4279:
4276:
4273:
4270:
4267:
4264:
4261:
4258:
4255:
4252:
4249:
4246:
4236:
4228:
4227:
4211:topologically
4209:
4198:
4193:
4189:
4185:
4182:
4179:
4176:
4173:
4170:
4167:
4164:
4161:
4158:
4155:
4152:
4149:
4146:
4143:
4140:
4137:
4127:
4119:
4118:
4112:
4109:
4097:
4094:
4091:
4088:
4085:
4082:
4079:
4076:
4073:
4070:
4067:
4064:
4061:
4056:
4052:
4048:
4045:
4042:
4039:
4036:
4026:
4019:
4015:
4014:
4012:
4001:
3996:
3992:
3988:
3985:
3982:
3979:
3974:
3970:
3966:
3961:
3957:
3953:
3950:
3947:
3944:
3941:
3931:
3923:
3918:
3917:
3915:
3904:
3899:
3895:
3891:
3888:
3885:
3882:
3877:
3873:
3869:
3864:
3860:
3856:
3853:
3850:
3847:
3844:
3834:
3826:
3821:
3820:
3818:
3807:
3802:
3798:
3794:
3791:
3788:
3785:
3780:
3776:
3772:
3767:
3763:
3759:
3756:
3753:
3750:
3747:
3737:
3729:
3724:
3723:
3721:
3710:
3705:
3701:
3697:
3694:
3691:
3688:
3683:
3679:
3675:
3670:
3666:
3662:
3659:
3656:
3653:
3650:
3640:
3632:
3628:
3623:
3622:
3610:
3607:
3602:
3597:
3593:
3581:
3580:
3579:
3563:
3560:
3557:
3553:
3547:
3543:
3537:
3534:
3531:
3527:
3523:
3518:
3514:
3508:
3505:
3502:
3498:
3492:
3488:
3477:
3465:
3462:
3459:
3456:
3453:
3447: if
3441:
3437:
3431:
3427:
3423:
3418:
3414:
3408:
3404:
3376:
3373:
3370:
3366:
3362:
3359:
3356:
3351:
3347:
3335:
3326:
3322:
3321:
3302:
3294:
3277:
3271:
3258:
3255:
3250:
3245:
3241:
3229:
3228:
3214:
3211:
3206:
3202:
3195:
3192:
3189:
3185:
3179:
3175:
3171:
3156:
3155:
3139:
3136:
3133:
3129:
3123:
3119:
3113:
3110:
3107:
3103:
3099:
3094:
3090:
3084:
3081:
3078:
3074:
3068:
3064:
3053:
3041:
3038:
3035:
3032:
3029:
3023: if
3017:
3013:
3007:
3003:
2999:
2994:
2990:
2984:
2980:
2969:
2957:
2954:
2949:
2944:
2940:
2914:
2911:
2908:
2904:
2900:
2897:
2894:
2889:
2885:
2873:
2859:
2855:
2854:
2852:
2829:
2826:
2823:
2820:
2817:
2807:
2796:
2795:
2793:
2782:
2779:
2776:
2773:
2770:
2767:
2764:
2759:
2755:
2751:
2746:
2742:
2738:
2735:
2732:
2729:
2726:
2716:
2692:
2691:
2689:
2678:
2675:
2672:
2669:
2666:
2663:
2660:
2657:
2654:
2651:
2648:
2638:
2628:
2627:
2619:
2612:
2601:
2596:
2593:
2589:
2585:
2580:
2577:
2573:
2569:
2566:
2563:
2558:
2554:
2550:
2545:
2541:
2537:
2532:
2529:
2525:
2521:
2518:
2515:
2512:
2509:
2499:
2497:dicyclic group
2490:
2486:
2485:
2483:
2472:
2467:
2463:
2459:
2456:
2453:
2450:
2445:
2441:
2437:
2434:
2431:
2428:
2425:
2415:
2408:
2404:
2403:
2392:
2381:
2376:
2372:
2368:
2365:
2362:
2359:
2354:
2350:
2346:
2341:
2337:
2333:
2330:
2327:
2324:
2321:
2311:
2305:dihedral group
2298:
2294:
2293:
2291:
2280:
2275:
2271:
2267:
2264:
2261:
2251:
2238:
2234:
2233:
2230:
2219:
2216:
2213:
2210:
2207:
2197:
2186:
2185:
2182:
2179:
2168:
2165:
2136:
2133:
2076:(respectively
2069:
2037:
2010:
2007:
1989:(respectively
1973:is finite and
1960:
1957:
1929:
1924:
1921:
1917:
1894:
1891:
1887:
1883:
1861:
1858:
1854:
1850:
1847:
1831:
1828:
1813:
1809:
1805:
1800:
1796:
1790:
1786:
1763:
1760:
1755:
1751:
1745:
1741:
1735:
1731:
1700:
1696:
1656:
1653:
1648:
1644:
1611:
1608:
1605:
1600:
1597:
1593:
1560:
1557:
1554:
1531:
1528:
1525:
1522:
1519:
1495:
1492:
1489:
1486:
1483:
1451:
1448:
1445:
1442:
1439:
1420:
1419:
1408:
1405:
1401:
1395:
1391:
1387:
1384:
1381:
1378:
1375:
1372:
1359:quotient group
1346:
1343:
1340:
1337:
1334:
1325:.) The group
1312:
1308:
1289:normal closure
1270:
1266:
1245:
1242:
1239:
1236:
1233:
1211:
1207:
1169:
1152:
1149:
1148:
1147:
1132:
1119:
1106:
1072:
1069:
1065:
1064:
1053:
1050:
1045:
1041:
1037:
1034:
1031:
1028:
1023:
1019:
1015:
1010:
1006:
1002:
999:
996:
993:
990:
960:
959:
948:
945:
942:
939:
934:
930:
926:
923:
920:
917:
912:
908:
904:
899:
895:
891:
888:
885:
882:
879:
827:
826:
815:
812:
809:
806:
801:
797:
793:
790:
787:
784:
781:
778:
773:
769:
765:
762:
759:
754:
750:
746:
743:
740:
737:
734:
719:
707:quotient group
702:
690:
681:
675:
669:
663:
657:
651:
618:
559:
558:
546:
537:
523:
478:
466:
463:dihedral group
419:
412:
405:
400:
399:
384:
380:
374:
370:
366:
359:
355:
349:
345:
337:
333:
327:
323:
295:
292:
274:
273:
262:
259:
254:
250:
246:
243:
240:
226:
225:
214:
211:
208:
205:
200:
196:
192:
189:
186:
159:the relations
121:
120:
109:
106:
103:
100:
97:
94:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5939:
5928:
5925:
5923:
5920:
5919:
5917:
5908:
5905:
5900:
5899:
5894:
5890:
5885:
5884:
5880:
5873:
5867:
5862:
5861:
5855:
5851:
5848:
5842:
5840:0-521-58542-2
5836:
5832:
5828:
5824:
5819:
5817:0-387-09212-9
5813:
5809:
5805:
5801:
5797:
5796:
5792:
5783:
5779:
5775:
5773:0-521-22729-1
5769:
5765:
5761:
5757:
5750:
5747:
5739:
5735:
5731:
5727:
5723:
5718:
5713:
5708:
5703:
5699:
5695:
5691:
5687:
5686:
5678:
5674:
5668:
5665:
5661:
5657:
5653:
5649:
5648:
5643:
5637:
5634:
5629:
5623:
5619:
5615:
5608:
5605:
5597:
5593:
5589:
5588:
5580:
5576:
5570:
5567:
5562:
5558:
5554:
5550:
5546:
5542:
5538:
5534:
5530:
5526:
5522:
5515:
5512:
5507:
5503:
5499:
5495:
5488:
5486:
5484:
5480:
5473:
5469:
5466:
5464:
5461:
5459:
5456:
5454:
5451:
5449:
5446:
5445:
5441:
5439:
5437:
5432:
5430:
5426:
5422:
5421:
5416:
5412:
5409:, called the
5408:
5404:
5400:
5394:
5388:
5382:
5374:
5372:
5370:
5366:
5362:
5358:
5354:
5350:
5346:
5342:
5338:
5331:
5325:
5316:
5312:
5306:
5298:
5293:
5289:
5285:
5279:
5275:
5271:
5267:
5260:
5256:
5252:
5251:
5246:
5241:
5237:
5233:
5229:
5222:
5218:
5214:
5213:
5208:
5207:
5206:
5204:
5200:
5194:
5190:
5184:
5178:
5174:
5168:
5161:Constructions
5160:
5158:
5156:
5155:William Boone
5152:
5151:Pyotr Novikov
5148:
5144:
5140:
5136:
5130:
5126:
5120:
5112:
5110:
5108:
5102:
5100:
5095:
5093:
5089:
5084:
5080:
5076:
5072:
5066:
5062:
5057:
5053:
5046:
5042:
5036:
5030:is normal in
5029:
5025:
5021:
5017:
5012:
5008:
5000:
4996:
4992:
4988:
4981:
4975:
4973:
4966:Some theorems
4965:
4963:
4962:with itself.
4961:
4940:
4928:
4924:
4915:
4911:
4892:
4882:
4879:
4875:
4871:
4866:
4858:
4855:
4846:
4841:
4833:
4830:
4827:
4824:
4821:
4815:
4810:
4802:
4799:
4796:
4790:
4785:
4777:
4774:
4768:
4763:
4759:
4755:
4750:
4746:
4742:
4739:
4736:
4733:
4723:
4721:
4718:
4717:
4714:
4695:
4692:
4688:
4682:
4678:
4674:
4671:
4666:
4662:
4658:
4655:
4652:
4649:
4639:
4637:
4633:
4629:
4625:
4624:
4621:
4604:
4601:
4598:
4595:
4592:
4589:
4586:
4583:
4580:
4577:
4574:
4571:
4566:
4563:
4559:
4553:
4550:
4546:
4542:
4539:
4536:
4533:
4530:
4527:
4524:
4521:
4518:
4515:
4505:
4503:
4500:
4499:
4496:
4492:
4488:
4484:
4480:
4476:
4472:
4453:
4449:
4445:
4440:
4436:
4432:
4429:
4426:
4423:
4413:
4411:
4410:modular group
4407:
4403:
4402:
4398:
4394:
4390:
4386:
4382:
4363:
4355:
4352:
4346:
4341:
4333:
4330:
4324:
4319:
4315:
4311:
4306:
4298:
4295:
4292:
4286:
4283:
4280:
4277:
4274:
4271:
4268:
4265:
4262:
4259:
4256:
4253:
4250:
4247:
4237:
4234:
4230:
4229:
4226:
4222:
4218:
4214:
4210:
4191:
4183:
4180:
4177:
4171:
4168:
4165:
4162:
4159:
4156:
4153:
4150:
4147:
4144:
4141:
4138:
4128:
4125:
4121:
4120:
4115:
4110:
4092:
4089:
4086:
4083:
4080:
4077:
4074:
4071:
4068:
4065:
4062:
4059:
4054:
4050:
4046:
4043:
4040:
4037:
4027:
4025:
4017:
4016:
4013:
3994:
3986:
3983:
3977:
3972:
3968:
3964:
3959:
3955:
3951:
3948:
3945:
3942:
3932:
3930:
3920:
3919:
3916:
3897:
3889:
3886:
3880:
3875:
3871:
3867:
3862:
3858:
3854:
3851:
3848:
3845:
3835:
3833:
3823:
3822:
3819:
3800:
3792:
3789:
3783:
3778:
3774:
3770:
3765:
3761:
3757:
3754:
3751:
3748:
3738:
3736:
3726:
3725:
3722:
3703:
3695:
3692:
3686:
3681:
3677:
3673:
3668:
3664:
3660:
3657:
3654:
3651:
3641:
3639:
3638:Klein 4 group
3625:
3624:
3608:
3605:
3600:
3595:
3591:
3582:
3561:
3558:
3555:
3551:
3545:
3541:
3535:
3532:
3529:
3525:
3521:
3516:
3512:
3506:
3503:
3500:
3496:
3490:
3486:
3478:
3463:
3460:
3457:
3454:
3451:
3439:
3435:
3429:
3425:
3421:
3416:
3412:
3406:
3402:
3394:
3393:
3392:
3374:
3371:
3368:
3364:
3360:
3357:
3354:
3349:
3345:
3336:
3334:
3329:
3324:
3323:
3319:
3315:
3311:
3305:
3301:
3297:
3293:
3289:
3285:
3280:
3276:
3272:
3270:
3256:
3253:
3248:
3243:
3239:
3212:
3209:
3204:
3193:
3190:
3187:
3183:
3177:
3173:
3161:
3160:
3159:
3137:
3134:
3131:
3127:
3121:
3117:
3111:
3108:
3105:
3101:
3097:
3092:
3088:
3082:
3079:
3076:
3072:
3066:
3062:
3054:
3039:
3036:
3033:
3030:
3027:
3015:
3011:
3005:
3001:
2997:
2992:
2988:
2982:
2978:
2970:
2955:
2952:
2947:
2942:
2938:
2930:
2929:
2912:
2909:
2906:
2902:
2898:
2895:
2892:
2887:
2883:
2874:
2871:
2867:
2862:
2857:
2856:
2853:
2851:
2847:
2843:
2824:
2821:
2818:
2808:
2806:
2802:
2798:
2797:
2794:
2777:
2774:
2771:
2768:
2765:
2762:
2757:
2753:
2749:
2744:
2740:
2736:
2733:
2730:
2727:
2717:
2715:
2712:
2708:
2704:
2701:
2697:
2694:
2693:
2690:
2673:
2670:
2667:
2664:
2661:
2658:
2655:
2652:
2649:
2639:
2637:
2633:
2630:
2629:
2625:
2617:
2613:
2594:
2591:
2587:
2583:
2578:
2575:
2571:
2567:
2564:
2561:
2556:
2552:
2548:
2543:
2539:
2535:
2530:
2527:
2523:
2519:
2516:
2513:
2510:
2500:
2498:
2493:
2488:
2487:
2484:
2465:
2457:
2454:
2448:
2443:
2439:
2435:
2432:
2429:
2426:
2416:
2414:
2406:
2405:
2402:a reflection
2401:
2397:
2393:
2374:
2366:
2363:
2357:
2352:
2348:
2344:
2339:
2335:
2331:
2328:
2325:
2322:
2312:
2310:
2306:
2301:
2296:
2295:
2292:
2273:
2269:
2265:
2262:
2252:
2250:
2246:
2241:
2236:
2235:
2231:
2211:
2208:
2198:
2196:
2192:
2188:
2187:
2183:
2180:
2177:
2176:
2173:
2166:
2164:
2162:
2158:
2155:, student of
2154:
2150:
2146:
2142:
2134:
2132:
2130:
2126:
2122:
2118:
2117:Graham Higman
2113:
2111:
2107:
2103:
2099:
2095:
2091:
2087:
2083:
2079:
2075:
2072:
2065:
2061:
2058:), calculate
2057:
2053:
2049:
2044:
2040:
2033:
2028:
2024:
2020:
2016:
2008:
2006:
2004:
2000:
1992:
1988:
1985:. A group is
1984:
1980:
1976:
1972:
1968:
1967:
1958:
1956:
1954:
1949:
1927:
1922:
1919:
1915:
1892:
1889:
1885:
1881:
1859:
1856:
1852:
1848:
1845:
1837:
1829:
1827:
1811:
1807:
1803:
1798:
1794:
1788:
1784:
1761:
1758:
1753:
1749:
1743:
1739:
1733:
1729:
1720:
1716:
1698:
1694:
1685:
1681:
1677:
1673:
1668:
1654:
1651:
1646:
1642:
1633:
1629:
1625:
1609:
1606:
1603:
1598:
1595:
1591:
1582:
1579:are words on
1578:
1574:
1558:
1555:
1552:
1543:
1526:
1523:
1520:
1509:
1490:
1487:
1484:
1473:
1469:
1465:
1446:
1443:
1440:
1429:
1425:
1406:
1403:
1399:
1393:
1389:
1385:
1379:
1376:
1373:
1363:
1362:
1361:
1360:
1341:
1338:
1335:
1310:
1306:
1297:
1293:
1290:
1286:
1268:
1264:
1240:
1237:
1234:
1209:
1205:
1196:
1192:
1188:
1184:
1180:
1176:
1172:
1165:
1158:
1150:
1143:
1139:
1133:
1129:
1125:
1120:
1116:
1112:
1107:
1102:
1098:
1092:
1091:
1090:
1085:
1081:
1070:
1068:
1051:
1043:
1035:
1032:
1026:
1021:
1017:
1013:
1008:
1004:
1000:
997:
994:
991:
981:
980:
979:
975:
971:
967:
946:
940:
937:
932:
924:
921:
915:
910:
906:
902:
897:
893:
889:
886:
883:
880:
870:
869:
868:
866:
860:
856:
852:
848:
841:
837:
833:
813:
807:
804:
799:
791:
788:
782:
779:
776:
771:
767:
763:
760:
757:
752:
748:
744:
741:
738:
735:
725:
724:
723:
716:
712:
708:
700:
696:
688:
684:
678:
672:
662:
656:
650:
646:
642:
638:
635:
631:
627:
622:
613:
609:
605:
601:
592:
588:
584:
580:
573:
569:
565:
551:
547:
542:
538:
533:
529:
528:
527:
521:
516:
512:
506:
502:
496:
491:
484:
476:
472:
464:
459:
457:
453:
449:
445:
441:
437:
432:
430:
426:
422:
415:
408:
382:
378:
372:
368:
364:
357:
353:
347:
343:
335:
331:
325:
321:
313:
312:
311:
309:
305:
301:
293:
291:
289:
284:
281:
279:
260:
252:
248:
244:
241:
231:
230:
229:
212:
206:
203:
198:
194:
190:
187:
177:
176:
175:
173:
169:
164:
162:
158:
154:
150:
146:
142:
138:
134:
130:
126:
107:
101:
98:
95:
85:
84:
83:
81:
77:
73:
69:
68:
63:
59:
55:
51:
47:
40:
33:
19:
5896:
5859:
5830:
5807:
5759:
5756:Wall, C.T.C.
5749:
5689:
5683:
5667:
5651:
5645:
5636:
5613:
5607:
5591:
5585:
5569:
5528:
5524:
5514:
5497:
5493:
5433:
5420:Bruhat order
5418:
5414:
5403:Cayley graph
5396:
5387:Cayley graph
5368:
5364:
5360:
5356:
5348:
5344:
5340:
5336:
5329:
5323:
5314:
5310:
5304:
5302:
5287:
5283:
5277:
5273:
5269:
5265:
5258:
5254:
5248:
5239:
5235:
5231:
5227:
5220:
5216:
5212:free product
5210:
5202:
5198:
5192:
5188:
5182:
5176:
5172:
5166:
5164:
5146:
5142:
5138:
5134:
5128:
5124:
5116:
5107:Cayley table
5104:
5098:
5097:
5091:
5087:
5082:
5078:
5074:
5070:
5060:
5055:
5048:
5044:
5040:
5031:
5027:
5019:
5015:
5010:
5003:
4990:
4983:
4979:
4977:
4971:
4970:
4920:
4631:
4627:
4494:
4490:
4486:
4482:
4479:free product
4474:
4405:
4396:
4388:
4384:
4232:
4216:
4212:
4123:
4113:
3390:
3337:generators:
3333:braid groups
3327:
3317:
3313:
3309:
3303:
3299:
3295:
3291:
3287:
3283:
3278:
3274:
3230:
3157:
2875:generators:
2869:
2860:
2849:
2841:
2804:
2713:
2710:
2706:
2702:
2699:
2695:
2635:
2631:
2623:
2491:
2399:
2395:
2308:
2299:
2248:
2245:cyclic group
2239:
2194:
2181:Presentation
2170:
2138:
2114:
2109:
2105:
2101:
2097:
2093:
2089:
2085:
2081:
2067:
2063:
2059:
2055:
2051:
2047:
2042:
2035:
2031:
2022:
2018:
2014:
2012:
2002:
1994:
1990:
1986:
1982:
1978:
1974:
1970:
1964:
1962:
1950:
1833:
1718:
1714:
1683:
1675:
1671:
1669:
1631:
1627:
1623:
1580:
1576:
1572:
1544:
1507:
1471:
1467:
1463:
1427:
1423:
1421:
1295:
1291:
1284:
1194:
1190:
1185:be a set of
1182:
1178:
1167:
1163:
1161:
1141:
1137:
1127:
1123:
1114:
1110:
1100:
1096:
1083:
1079:
1074:
1066:
973:
969:
965:
961:
864:
858:
854:
850:
846:
839:
835:
831:
828:
714:
710:
698:
694:
686:
679:
673:
667:
660:
654:
648:
644:
640:
636:
633:
629:
625:
623:
611:
607:
603:
599:
590:
586:
582:
578:
571:
567:
563:
560:
549:
540:
531:
519:
514:
510:
504:
500:
497:
489:
482:
474:
470:
460:
455:
451:
447:
443:
439:
435:
433:
428:
424:
417:
410:
403:
401:
307:
303:
297:
285:
282:
277:
275:
227:
171:
168:cyclic group
165:
160:
152:
136:
132:
128:
124:
123:Informally,
122:
79:
75:
71:
65:
61:
57:
50:presentation
49:
43:
5500:(1): 3–10.
5411:word metric
5393:Word metric
4383:nontrivial
4221:Dehn twists
3391:relations:
2928:relations:
2846:commutators
2157:Felix Klein
2129:uncountably
2112:recursive.
46:mathematics
5916:Categories
5793:References
5782:0423.20029
5734:0086.24701
5660:0068.01301
5415:weak order
5337:deficiency
5305:deficiency
5299:Deficiency
5292:commutator
5099:Corollary.
4914:commutator
4720:Tits group
2307:of order 2
2191:free group
1470:. A group
1428:generators
1175:free group
1151:Definition
576:, and let
520:not unique
402:where the
300:free group
294:Background
149:free group
141:isomorphic
67:generators
5898:MathWorld
5654:: 1–143,
5561:120100270
5553:0080-4630
5369:efficient
5157:in 1958.
5002:φ :
4993:. By the
4941:≀
4898:⟩
4880:−
4743:∣
4731:⟨
4701:⟩
4693:−
4659:∣
4647:⟨
4608:⟩
4564:−
4551:−
4531:∣
4513:⟨
4477:) is the
4459:⟩
4433:∣
4421:⟨
4395:of SL(2,
4369:⟩
4263:∣
4245:⟨
4197:⟩
4148:∣
4136:⟨
4096:⟩
4047:∣
4035:⟨
4000:⟩
3952:∣
3940:⟨
3903:⟩
3855:∣
3843:⟨
3806:⟩
3758:∣
3746:⟨
3709:⟩
3661:∣
3649:⟨
3592:σ
3552:σ
3542:σ
3526:σ
3513:σ
3497:σ
3487:σ
3461:±
3455:≠
3436:σ
3426:σ
3413:σ
3403:σ
3372:−
3365:σ
3358:…
3346:σ
3240:σ
3184:σ
3174:σ
3128:σ
3118:σ
3102:σ
3089:σ
3073:σ
3063:σ
3037:±
3031:≠
3012:σ
3002:σ
2989:σ
2979:σ
2939:σ
2910:−
2903:σ
2896:…
2884:σ
2828:⟩
2822:∣
2816:⟨
2781:⟩
2737:∣
2725:⟨
2677:⟩
2659:∣
2647:⟨
2600:⟩
2592:−
2576:−
2520:∣
2508:⟨
2471:⟩
2436:∣
2424:⟨
2380:⟩
2332:∣
2320:⟨
2279:⟩
2266:∣
2260:⟨
2247:of order
2218:⟩
2215:∅
2212:∣
2206:⟨
2184:Comments
2121:countably
2074:recursive
1940:for some
1920:−
1890:−
1857:−
1849:∪
1759:−
1678:from the
1607:∈
1596:−
1530:⟩
1524:∣
1518:⟨
1494:⟩
1488:∣
1482:⟨
1450:⟩
1444:∣
1438:⟨
1383:⟩
1377:∣
1371:⟨
1345:⟩
1339:∣
1333:⟨
1244:⟩
1238:∣
1232:⟨
1049:⟩
1001:∣
989:⟨
944:⟩
890:∣
878:⟨
811:⟩
745:∣
733:⟨
365:⋯
302:on a set
258:⟩
245:∣
239:⟨
210:⟩
191:∣
185:⟨
170:of order
105:⟩
99:∣
93:⟨
76:relations
5856:(1994).
5829:(1997).
5806:(1980).
5738:archived
5726:16590307
5675:(1958),
5596:Archived
5577:(1856).
5442:See also
5417:and the
5335:and the
5319:is just
5165:Suppose
4972:Theorem.
4960:integers
2872:symbols
2167:Examples
2034: :
1776:, where
1468:relators
1144:⟩
1136:⟨
1103:⟩
1095:⟨
1086:⟩
1078:⟨
1071:Notation
701:. Thus D
452:uniquely
308:uniquely
278:relators
145:quotient
5758:(ed.).
5694:Bibcode
5594:: 446.
5533:Bibcode
5077:⟩ ≅ im(
5022:be the
4912:is the
4634:), the
4473:PSL(2,
4408:), the
4404:PSL(2,
4223:on the
2135:History
1181:. Let
1173:be the
1099:|
1082:|
615:
594:
553:
155:by the
143:to the
5868:
5849:, etc.
5837:
5814:
5780:
5770:
5732:
5724:
5717:528693
5714:
5658:
5624:
5559:
5551:
5407:metric
5333:|
5321:|
5024:kernel
4231:GL(2,
4122:SL(2,
4022:, the
3927:, the
3830:, the
3733:, the
3636:, the
3567:
3331:, the
3231:using
3216:
3143:
2864:, the
2840:where
2495:, the
2411:, the
2303:, the
2243:, the
1571:where
861:) = 1}
857:= 1, (
842:
488:s and
5741:(PDF)
5680:(PDF)
5599:(PDF)
5582:(PDF)
5557:S2CID
5474:Notes
5290:(cf.
5197:with
4225:torus
3922:I ≅ A
3825:O ≅ S
3728:T ≅ A
3320:+2}.
3273:Here
2394:Here
2178:Group
2080:) if
1193:, so
1187:words
853:= 1,
493:'
486:'
147:of a
54:group
5866:ISBN
5835:ISBN
5812:ISBN
5768:ISBN
5722:PMID
5622:ISBN
5549:ISSN
5303:The
5247:the
5209:the
5201:and
5181:and
5145:and
5090:and
5081:) =
5047:⟩ =
4489:and
4215:and
3316:+1,
2799:the
2626:= 2
2614:The
2189:the
1717:and
1626:and
1575:and
1162:Let
666:...
604:rfrf
583:rfrf
545:, or
532:rfrf
522:in D
48:, a
5778:Zbl
5730:Zbl
5712:PMC
5702:doi
5656:Zbl
5618:374
5541:doi
5529:262
5502:doi
5367:is
5280:, ⟩
5244:and
4989:on
4626:BS(
3631:≅ D
2868:on
2803:on
2489:Dic
2193:on
2066:of
2013:If
1977:if
1969:if
1944:in
1907:or
1713:of
1506:if
1430:of
1298:in
1294:of
1189:on
1177:on
972:, (
849:= {
834:= {
639:of
581:= ⟨
566:= ⟨
555:= 1
543:= 1
534:= 1
501:rfr
495:s.
434:If
151:on
74:of
64:of
44:In
5918::
5895:.
5891:.
5802:;
5776:.
5736:,
5728:,
5720:,
5710:,
5700:,
5690:44
5688:,
5682:,
5652:44
5620:.
5592:12
5590:.
5584:.
5555:.
5547:.
5539:.
5527:.
5523:.
5498:70
5496:.
5482:^
5431:.
5313:|
5294:).
5276:,
5272:|
5268:,
5257:×
5238:,
5234:|
5230:,
5219:∗
5191:|
5175:|
5137:,
5127:|
5073:|
5067:,
5043:|
5009:→
4786:13
4630:,
4493:/3
4485:/2
4399:)
4391:–
4387:/2
4235:)
4126:)
3621:.
3312:,
3306:+1
3269:.
2705:×
2634:×
2163:.
2041:→
2029:)
2005:.
1993:,
1955:.
1667:.
1542:.
1140:;
1126:;
1113:|
976:)}
974:rf
968:,
859:rf
838:,
637:Rx
621:.
610:,
606:,
589:,
585:,
570:,
513:=
508:,
503:=
458:.
298:A
290:.
163:.
5901:.
5874:.
5843:.
5820:.
5784:.
5704::
5696::
5630:.
5563:.
5543::
5535::
5508:.
5504::
5365:G
5361:G
5357:G
5349:G
5345:G
5341:G
5330:R
5324:S
5317:⟩
5315:R
5311:S
5309:⟨
5288:T
5284:S
5278:Q
5274:R
5270:T
5266:S
5264:⟨
5259:H
5255:G
5242:⟩
5240:Q
5236:R
5232:T
5228:S
5226:⟨
5221:H
5217:G
5203:T
5199:S
5195:⟩
5193:Q
5189:T
5187:⟨
5183:H
5179:⟩
5177:R
5173:S
5171:⟨
5167:G
5147:v
5143:u
5139:v
5135:u
5131:⟩
5129:R
5125:S
5123:⟨
5092:K
5088:G
5083:G
5079:φ
5075:K
5071:G
5069:⟨
5061:φ
5056:K
5054:/
5051:G
5049:F
5045:K
5041:G
5039:⟨
5034:G
5032:F
5028:K
5020:K
5016:G
5011:G
5006:G
5004:F
4991:G
4986:G
4984:F
4980:G
4945:Z
4937:Z
4893:6
4889:)
4883:1
4876:b
4872:a
4867:4
4863:)
4859:b
4856:a
4853:(
4850:(
4847:,
4842:4
4838:]
4834:b
4831:a
4828:b
4825:,
4822:a
4819:[
4816:,
4811:5
4807:]
4803:b
4800:,
4797:a
4794:[
4791:,
4782:)
4778:b
4775:a
4772:(
4769:,
4764:3
4760:b
4756:,
4751:2
4747:a
4740:b
4737:,
4734:a
4696:1
4689:b
4683:m
4679:a
4675:b
4672:=
4667:n
4663:a
4656:b
4653:,
4650:a
4632:n
4628:m
4605:y
4602:z
4599:=
4596:z
4593:y
4590:,
4587:x
4584:z
4581:=
4578:z
4575:x
4572:,
4567:1
4560:y
4554:1
4547:x
4543:y
4540:x
4537:=
4534:z
4528:z
4525:,
4522:y
4519:,
4516:x
4495:Z
4491:Z
4487:Z
4483:Z
4475:Z
4454:3
4450:b
4446:,
4441:2
4437:a
4430:b
4427:,
4424:a
4406:Z
4397:Z
4389:Z
4385:Z
4364:2
4360:)
4356:b
4353:j
4350:(
4347:,
4342:2
4338:)
4334:a
4331:j
4328:(
4325:,
4320:2
4316:j
4312:,
4307:4
4303:)
4299:a
4296:b
4293:a
4290:(
4287:,
4284:b
4281:a
4278:b
4275:=
4272:a
4269:b
4266:a
4260:j
4257:,
4254:b
4251:,
4248:a
4233:Z
4217:b
4213:a
4192:4
4188:)
4184:a
4181:b
4178:a
4175:(
4172:,
4169:b
4166:a
4163:b
4160:=
4157:a
4154:b
4151:a
4145:b
4142:,
4139:a
4124:Z
4114:n
4093:j
4090:=
4087:i
4084:j
4081:i
4078:,
4075:i
4072:=
4069:j
4066:i
4063:j
4060:,
4055:4
4051:i
4044:j
4041:,
4038:i
4020:8
4018:Q
3995:5
3991:)
3987:t
3984:s
3981:(
3978:,
3973:3
3969:t
3965:,
3960:2
3956:s
3949:t
3946:,
3943:s
3924:5
3898:4
3894:)
3890:t
3887:s
3884:(
3881:,
3876:3
3872:t
3868:,
3863:2
3859:s
3852:t
3849:,
3846:s
3827:4
3801:3
3797:)
3793:t
3790:s
3787:(
3784:,
3779:3
3775:t
3771:,
3766:2
3762:s
3755:t
3752:,
3749:s
3730:4
3704:2
3700:)
3696:t
3693:s
3690:(
3687:,
3682:2
3678:t
3674:,
3669:2
3665:s
3658:t
3655:,
3652:s
3633:2
3629:4
3627:V
3609:1
3606:=
3601:2
3596:i
3562:1
3559:+
3556:i
3546:i
3536:1
3533:+
3530:i
3522:=
3517:i
3507:1
3504:+
3501:i
3491:i
3476:,
3464:1
3458:i
3452:j
3440:i
3430:j
3422:=
3417:j
3407:i
3375:1
3369:n
3361:,
3355:,
3350:1
3328:n
3325:B
3318:i
3314:i
3310:i
3304:i
3300:σ
3296:i
3292:σ
3288:i
3284:i
3279:i
3275:σ
3257:1
3254:=
3249:2
3244:i
3213:1
3210:=
3205:3
3201:)
3194:1
3191:+
3188:i
3178:i
3170:(
3138:1
3135:+
3132:i
3122:i
3112:1
3109:+
3106:i
3098:=
3093:i
3083:1
3080:+
3077:i
3067:i
3052:,
3040:1
3034:i
3028:j
3016:i
3006:j
2998:=
2993:j
2983:i
2968:,
2956:1
2953:=
2948:2
2943:i
2913:1
2907:n
2899:,
2893:,
2888:1
2870:n
2861:n
2858:S
2850:S
2842:R
2825:R
2819:S
2805:S
2778:x
2775:y
2772:=
2769:y
2766:x
2763:,
2758:n
2754:y
2750:,
2745:m
2741:x
2734:y
2731:,
2728:x
2714:Z
2711:n
2709:/
2707:Z
2703:Z
2700:m
2698:/
2696:Z
2674:x
2671:y
2668:=
2665:y
2662:x
2656:y
2653:,
2650:x
2636:Z
2632:Z
2624:n
2620:8
2618:Q
2595:1
2588:r
2584:=
2579:1
2572:f
2568:r
2565:f
2562:,
2557:2
2553:f
2549:=
2544:n
2540:r
2536:,
2531:n
2528:2
2524:r
2517:f
2514:,
2511:r
2492:n
2466:2
2462:)
2458:f
2455:r
2452:(
2449:,
2444:2
2440:f
2433:f
2430:,
2427:r
2409:∞
2407:D
2400:f
2396:r
2375:2
2371:)
2367:f
2364:r
2361:(
2358:,
2353:2
2349:f
2345:,
2340:n
2336:r
2329:f
2326:,
2323:r
2309:n
2300:n
2297:D
2274:n
2270:a
2263:a
2249:n
2240:n
2237:C
2209:S
2195:S
2110:R
2106:R
2094:R
2090:S
2086:U
2084:(
2082:f
2070:S
2068:F
2064:U
2060:w
2056:w
2054:(
2052:f
2048:S
2043:N
2038:S
2036:F
2032:f
2023:N
2019:I
2015:S
1979:R
1971:S
1946:S
1942:x
1928:x
1923:1
1916:x
1893:1
1886:x
1882:x
1860:1
1853:S
1846:S
1812:k
1808:g
1804:=
1799:j
1795:g
1789:i
1785:g
1762:1
1754:k
1750:g
1744:j
1740:g
1734:i
1730:g
1719:R
1715:G
1699:i
1695:g
1684:S
1676:G
1672:G
1655:1
1652:=
1647:n
1643:r
1632:r
1628:y
1624:x
1610:R
1604:x
1599:1
1592:y
1581:S
1577:y
1573:x
1559:y
1556:=
1553:x
1527:R
1521:S
1508:G
1491:R
1485:S
1472:G
1464:R
1447:R
1441:S
1424:S
1407:.
1404:N
1400:/
1394:S
1390:F
1386:=
1380:R
1374:S
1342:R
1336:S
1311:S
1307:F
1296:R
1292:N
1285:R
1269:S
1265:F
1241:R
1235:S
1210:S
1206:F
1195:R
1191:S
1183:R
1179:S
1170:S
1168:F
1164:S
1159:.
1142:R
1138:S
1131:}
1128:R
1124:S
1122:{
1117:)
1115:R
1111:S
1109:(
1101:R
1097:S
1084:R
1080:S
1052:.
1044:2
1040:)
1036:f
1033:r
1030:(
1027:,
1022:2
1018:f
1014:,
1009:8
1005:r
998:f
995:,
992:r
970:f
966:r
964:{
947:.
941:1
938:=
933:2
929:)
925:f
922:r
919:(
916:=
911:2
907:f
903:=
898:8
894:r
887:f
884:,
881:r
865:R
855:f
851:r
847:R
840:f
836:r
832:S
814:.
808:1
805:=
800:2
796:)
792:f
789:r
786:(
783:,
780:1
777:=
772:2
768:f
764:,
761:1
758:=
753:8
749:r
742:f
739:,
736:r
720:8
715:N
713:/
711:F
703:8
699:F
695:N
691:8
687:N
682:m
680:x
676:m
674:r
670:m
668:x
664:1
661:x
658:1
655:r
652:1
649:x
645:N
641:R
634:x
630:F
626:N
619:8
612:f
608:r
600:R
596:⟩
591:f
587:r
579:R
574:⟩
572:f
568:r
564:F
557:.
550:f
541:r
536:,
524:8
515:r
511:r
505:f
490:f
483:r
479:8
475:f
471:r
467:8
465:D
456:G
448:G
444:G
440:S
436:G
425:n
420:i
418:a
413:i
411:s
406:i
404:s
383:n
379:a
373:n
369:s
358:2
354:a
348:2
344:s
336:1
332:a
326:1
322:s
304:S
261:,
253:n
249:a
242:a
213:,
207:1
204:=
199:n
195:a
188:a
172:n
161:R
153:S
137:G
133:R
129:S
125:G
108:.
102:R
96:S
80:G
72:R
62:S
58:G
41:.
34:.
20:)
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