3094:
2778:
38:
3089:{\displaystyle {\begin{aligned}{\begin{bmatrix}a&0\\b&1\end{bmatrix}}H=&~\left\{{\begin{bmatrix}a&0\\b&1\end{bmatrix}}{\begin{bmatrix}1&0\\c&1\end{bmatrix}}\colon c\in \mathbb {R} \right\}\\=&~\left\{{\begin{bmatrix}a&0\\b+c&1\end{bmatrix}}\colon c\in \mathbb {R} \right\}\\=&~\left\{{\begin{bmatrix}a&0\\d&1\end{bmatrix}}\colon d\in \mathbb {R} \right\}.\end{aligned}}}
3674:. In the event that the transmission errors occurred in precisely the non-zero positions of the coset leader the result will be the right codeword. In this example, if a single error occurs, the method will always correct it, since all possible coset leaders with a single one appear in the array.
2675:
3201:
2769:
2485:
3669:
The decoding procedure is to find the received word in the table and then add to it the coset leader of the row it is in. Since in binary arithmetic adding is the same operation as subtracting, this always results in an element of
946:
have now appeared in one of these cosets, generating any more can not give new cosets; any new coset would have to have an element in common with one of these and therefore would be identical to one of these cosets. For instance,
392:
varies through the group, it would appear that many cosets (right or left) would be generated. Nevertheless, it turns out that any two left cosets (respectively right cosets) are either disjoint or are identical as sets.
3391:(what we now call left and right cosets). If these decompositions coincided, that is, if the left cosets are the same as the right cosets, then there was a way to reduce the problem to one of working over
2576:
3528:
and there may be some choice in selecting it. Now the process is repeated, a new vector with a minimal number of ones that does not already appear is selected as a new coset leader and the coset of
3111:
2783:
2688:
3335:'s work of 1830β31. He introduced a notation but did not provide a name for the concept. The term "co-set" apparently appears for the first time in 1910 in a paper by G. A. Miller in the
443:} (see below for a extension to right cosets and double cosets). However, some authors (including Dummit & Foote and Rotman) reserve this notation specifically for representing the
77:
2418:
3763:, every vector in the same coset will have the same syndrome. To decode, the search is now reduced to finding the coset leader that has the same syndrome as the received word.
3492:
and if only a few errors are made in transmission it can be done effectively with only a very few mistakes. One method used for decoding uses an arrangement of the elements of
3337:
1960:
3403:
in his commentaries on Galois's work in 1865 and 1869 elaborated on these ideas and defined normal subgroups as we have above, although he did not use this term.
1497:. There are other types of equivalence relations in a group, such as conjugacy, that form different classes which do not have the properties discussed here.
3484:) is transmitted some of its bits may be altered in the process and the task of the receiver is to determine the most likely codeword that the corrupted
3680:
can be used to improve the efficiency of this method. It is a method of computing the correct coset (row) that a received word will be in. For an
4702:
4646:
4619:
4511:
4186:
1883:
226:
3520:
containing this element is written as the second row (namely, the row is formed by taking the sum of this element with each element of
4752:
4722:
4684:
4666:
4601:
4583:
3362:
4319:
4834:
1490:
4829:
1494:
2670:{\displaystyle G=\left\{{\begin{bmatrix}a&0\\b&1\end{bmatrix}}\colon a,b\in \mathbb {R} ,a\neq 0\right\},}
471:
47:
4850:
4238:
4197:
1901:
1829:. Since every right coset is a left coset, there is no need to distinguish "left cosets" from "right cosets".
3196:{\displaystyle T=\left\{{\begin{bmatrix}a&0\\0&1\end{bmatrix}}\colon a\in \mathbb {R} -\{0\}\right\}}
4179:
3346:
1868:
itself. Furthermore, the number of left cosets is equal to the number of right cosets and is known as the
1482:
280:
whose operation is written multiplicatively (juxtaposition denotes the group operation). Given an element
4877:
3476:
is a subgroup of this group. Codes can be used to correct errors that can occur in transmission. When a
2764:{\displaystyle H=\left\{{\begin{bmatrix}1&0\\c&1\end{bmatrix}}\colon c\in \mathbb {R} \right\}.}
1772:
261:
3516:
with a minimal number of ones that does not already appear in the top row is selected and the coset of
4824:
3976:. Several authors working with these sets have developed a specialized notation for their work, where
3508:
put into tabular form in a certain way. Namely, the top row of the array consists of the elements of
3373:
3308:
1431:
4218:
3693:
3369:
2496:
2115:
is a normal subgroup. (The same argument shows that every subgroup of an
Abelian group is normal.)
1838:
1474:
211:
148:
125:
2480:{\displaystyle \{\mathbf {x} \in V\mid \mathbf {x} =\mathbf {a} +\mathbf {w} ,\mathbf {w} \in W\}}
4553:
4535:
4264:
4172:
3512:, written in any order, except that the zero vector should be written first. Then, an element of
2315:
1580:
4805:
4786:
4748:
4718:
4698:
4680:
4662:
4642:
4615:
4597:
4579:
4507:
4494:
Seitz, Gary M. (1998), "Double Cosets in
Algebraic Groups", in Carter, R.W.; Saxl, J. (eds.),
4269:
4242:
3677:
2499:
to the subspace, which is a line or plane going through the origin. For example, consider the
2491:, and are cosets (both left and right, since the group is abelian). In terms of 3-dimensional
155:
80:
4742:
4712:
4656:
4545:
4526:
Duckworth, W. Ethan (2004), "Infiniteness of double coset collections in algebraic groups",
4499:
3354:
2500:
37:
4738:
2492:
2488:
2384:
2380:
1702:
1634:
249:
4190:
225:
Cosets are a basic tool in the study of groups; for example, they play a central role in
3728:
3501:
3443:
3400:
2355:
1982:
1680:
1493:
in the equivalence class sense. A set of representatives of all the cosets is called a
444:
253:
117:
4871:
2376:
1510:
397:
396:
If the group operation is written additively, as is often the case when the group is
163:
4557:
3422:, while most common today, has not been universally true in the past. For instance,
4549:
4323:
3772:
3525:
3230:
3216:
2372:
533:
257:
230:
137:
4503:
1711:, then its left cosets are different from its right cosets. That is, there is an
4808:
3381:
3341:(vol. 41, p. 382). Various other terms have been used including the German
2084:
1850:
183:
133:
4789:
4445:
The transpose matrix is used so that the vectors can be written as row vectors.
4844:
4168:
4813:
4794:
4775:
3760:
2388:
1962:
This equation can be generalized to the case where the groups are infinite.
194:
itself is both a left coset and a right coset. The number of left cosets of
4770:
17:
3885:
are either disjoint or identical. The set of all double cosets for fixed
1854:
1313:
The disjointness of non-identical cosets is a result of the fact that if
141:
3532:
containing it is the next row. The process ends when all the vectors of
1800:
is normal the set of all cosets forms a group called the quotient group
31:
4856:
4540:
3332:
167:
36:
4185:
Cosets are important in computational group theory. For example,
1430:
Two elements being in the same left coset also provide a natural
1427:
is itself a left coset (and the one that contains the identity).
182:. Cosets (both left and right) have the same number of elements (
2495:
vectors, these affine subspaces are all the "lines" or "planes"
4840:
1653:
the corresponding left and right cosets are equal, that is,
3319:, while the orbit under the left action is the right coset
256:. Cosets also appear in other areas of mathematics such as
3539:
An example of a standard array for the 2-dimensional code
3357:). (Note that Miller abbreviated his self-citation to the
3376:. A tool that he developed was in noting that a subgroup
3096:
That is, the left cosets consist of all the matrices in
252:) can be used as the elements of another group called a
27:
Disjoint, equal-size subsets of a group's underlying set
1667:
in the first example above. Furthermore, the cosets of
3131:
3029:
2951:
2879:
2843:
2791:
2708:
2596:
1481:. As with any set of equivalence classes, they form a
3114:
2781:
2691:
2579:
2421:
1904:
300:
are the sets obtained by multiplying each element of
87:
contains only 0 and 4. There are four left cosets of
50:
4695:
A First Course in
Abstract Algebra with Applications
3488:
could have started out as. This procedure is called
2371:
Another example of a coset comes from the theory of
2375:. The elements (vectors) of a vector space form an
532:. This is enough information to fill in the entire
116:(written using additive notation since this is the
3195:
3088:
2763:
2669:
2479:
1954:
1415:belongs to exactly one left coset of the subgroup
71:
4655:Joshi, K. D. (1989), "Β§5.2 Cosets of Subgroups",
3338:Quarterly Journal of Pure and Applied Mathematics
2341:, and so, can be used to form the quotient group
1886:allows us to compute the index in the case where
3368:Galois was concerned with deciding when a given
3100:having the same upper-left entry. This subgroup
1446:, to be equivalent with respect to the subgroup
431:is sometimes used for the set of (left) cosets {
3504:. A standard array is a coset decomposition of
3434:, emphasizing the subgroup being on the right.
3365:, which did not start publication until 1930.)
1747:is a different partition than the partition of
352:are defined similarly, except that the element
4747:, Courier Dover Publications, pp. 10 ff,
4717:, Courier Dover Publications, pp. 19 ff,
4711:Scott, W.R. (1987), "Β§1.7 Cosets and index",
3524:directly above it). This element is called a
248:. Cosets of a particular type of subgroup (a
8:
4287:
4285:
4178:Cosets are central in the definition of the
3185:
3179:
2474:
2422:
120:). Together they partition the entire group
4417:
4415:
2118:This example may be generalized. Again let
1200:. The following statements are equivalent:
316:is the left factor). In symbols these are,
236:, the number of elements of every subgroup
124:into equal-size, non-overlapping sets. The
4661:, New Age International, pp. 322 ff,
4366:
4364:
2168:is a positive integer. Then the cosets of
1500:Similar statements apply to right cosets.
202:is equal to the number of right cosets of
72:{\displaystyle \mathbb {Z} /8\mathbb {Z} }
4539:
4496:Algebraic Groups and their Representation
3496:(a received word could be any element of
3315:under the right action is the left coset
3172:
3171:
3126:
3113:
3070:
3069:
3024:
2998:
2997:
2946:
2920:
2919:
2874:
2838:
2786:
2782:
2780:
2749:
2748:
2703:
2690:
2643:
2642:
2591:
2578:
2573:be the multiplicative group of matrices,
2463:
2455:
2447:
2439:
2425:
2420:
1944:
1936:
1913:
1905:
1903:
447:formed from the cosets in the case where
65:
64:
56:
52:
51:
49:
4677:Theory and Applications of Finite Groups
4421:
4370:
4355:
3789:(which need not be distinct) of a group
3552:
2079:, so there are no other right cosets of
1477:of this relation are the left cosets of
538:
154:may be used to decompose the underlying
4477:
4475:
4281:
2124:be the additive group of the integers,
2004:, +) = ({..., β6, β3, 0, 3, 6, ...}, +)
1553:. For general groups, given an element
1055:, no left coset is also a right coset.
4454:
4433:
4406:
4394:
4343:
4302:
4300:
4291:
3906:contains the complete right cosets of
1759:. This is illustrated by the subgroup
4466:
4382:
3331:The concept of a coset dates back to
2073:. These three sets partition the set
1767:cosets may coincide. For example, if
474:. Its elements may be represented by
7:
4481:
4306:
3692:-dimensional binary vector space, a
3423:
1849:has the same number of elements (or
1663:. This is the case for the subgroup
1159:. In this case, every left coset of
4658:Foundations of Discrete Mathematics
2520:is a subgroup of the abelian group
1794:On the other hand, if the subgroup
1735:. This means that the partition of
1392:is a group, left multiplication by
4612:A First Course in Abstract Algebra
3536:have been sorted into the cosets.
2391:of this group. For a vector space
2129:= ({..., β2, β1, 0, 1, 2, ...}, +)
1990:= ({..., β2, β1, 0, 1, 2, ...}, +)
244:divides the number of elements of
210:. This common value is called the
25:
4073:denotes the set of double cosets
3550:(with 32 vectors) is as follows:
3438:An application from coding theory
4614:(5th ed.), Addison-Wesley,
4167:are used in the construction of
4029:denotes the set of right cosets
3928:and the complete left cosets of
3359:Quarterly Journal of Mathematics
2464:
2456:
2448:
2440:
2426:
1643:if and only if for all elements
872:. The (distinct) left cosets of
356:is now a right factor, that is,
4697:(3rd ed.), Prentice-Hall,
4123:denotes the double coset space
3989:denotes the set of left cosets
3844:. These are the left cosets of
2107:. That is, every left coset of
400:, the notation used changes to
4550:10.1016/j.jalgebra.2003.08.011
4498:, Springer, pp. 241β257,
3544:= {00000, 01101, 10110, 11011}
3472:is an additive abelian group,
3387:induced two decompositions of
1945:
1937:
1933:
1921:
1914:
1906:
1681:quotient group or factor group
1579:is also the left coset of the
254:quotient group or factor group
1:
3361:; this does not refer to the
2512:is a line through the origin
1843:Every left or right coset of
1763:in the first example above. (
83:under addition. The subgroup
4863:. The Group Properties Wiki.
4628:Hall, Jr., Marshall (1959),
4504:10.1007/978-94-011-5308-9_13
1051:In this example, except for
222:and is usually denoted by .
4830:Encyclopedia of Mathematics
4578:, Wm. C. Brown Publishers,
4231:is a discrete subgroup (of
4107:, sometimes referred to as
3546:in the 5-dimensional space
3448:A binary linear code is an
3211:As orbits of a group action
472:dihedral group of order six
276:be a subgroup of the group
4894:
4741:(1999), "Β§1.4 Subgroups",
4693:Rotman, Joseph J. (2006),
4637:Jacobson, Nathan (2009) ,
4610:Fraleigh, John B. (1994),
4594:Classical Abstract Algebra
4227:is a closed subgroup, and
4187:Thistlethwaite's algorithm
3964:be a group with subgroups
3770:
3460:-dimensional vector space
3441:
3214:
2111:is also a right coset, so
1836:
942:Since all the elements of
29:
4592:Dean, Richard A. (1990),
4574:Burton, David M. (1988),
4193:relies heavily on cosets.
3817:are the sets of the form
3715:having the property that
3229:can be used to define an
2267:. There are no more than
1485:of the underlying set. A
1434:. Define two elements of
1343:then there must exist an
1174:be a subgroup of a group
1163:is also a right coset of
229:that states that for any
4534:(2), Elsevier: 718β733,
4239:properly discontinuously
4200:is a double coset space
3363:journal of the same name
2775:consider the left coset
2387:of the vector space are
1955:{\displaystyle |G|=|H|.}
1741:into the left cosets of
1679:form a group called the
30:Not to be confused with
4823:Ivanova, O.A. (2001) ,
4675:Miller, G. A. (2012) ,
4641:(2nd ed.), Dover,
4632:, The Macmillan Company
3241:in two natural ways. A
2771:For a fixed element of
3464:over the binary field
3452:-dimensional subspace
3197:
3090:
2765:
2671:
2481:
1956:
1409:Thus every element of
1111:. The right cosets of
262:error-correcting codes
129:
73:
3777:Given two subgroups,
3198:
3091:
2766:
2672:
2482:
2403:, and a fixed vector
2006:. Then the cosets of
1957:
1753:into right cosets of
1721:such that no element
1569:, the right coset of
1078:. The left cosets of
74:
40:
4851:Illustrated examples
4744:The Theory of Groups
4630:The Theory of Groups
3893:form a partition of
3850:and right cosets of
3374:solvable by radicals
3112:
2779:
2689:
2577:
2557:and passing through
2419:
1902:
1487:coset representative
1460:(or equivalently if
1432:equivalence relation
1396:is a bijection, and
968:The right cosets of
48:
4739:Zassenhaus, Hans J.
4679:, Applewood Books,
4219:reductive Lie group
4198:CliffordβKlein form
3694:parity check matrix
3370:polynomial equation
3108:, but the subgroup
2018:are the three sets
1839:Index of a subgroup
1833:Index of a subgroup
1810:with the operation
1531:for every subgroup
1475:equivalence classes
304:by a fixed element
4825:"Coset in a group"
4806:Weisstein, Eric W.
4787:Weisstein, Eric W.
4596:, Harper and Row,
4528:Journal of Algebra
4173:non-measurable set
4109:double coset space
3873:Two double cosets
3684:-dimensional code
3406:Calling the coset
3193:
3156:
3086:
3084:
3054:
2982:
2904:
2868:
2816:
2761:
2733:
2667:
2621:
2477:
1952:
1884:Lagrange's theorem
1853:in the case of an
1581:conjugate subgroup
1543:and every element
1386:. Moreover, since
227:Lagrange's theorem
130:
69:
4704:978-0-13-186267-8
4648:978-0-486-47189-1
4621:978-0-201-53467-2
4513:978-0-7923-5292-1
4270:Coset enumeration
4243:homogeneous space
4150:More applications
3678:Syndrome decoding
3667:
3666:
3203:is not normal in
3018:
2940:
2832:
2677:and the subgroup
2536:, then the coset
1985:of the integers,
1178:and suppose that
854:
853:
505:. In this group,
16:(Redirected from
4885:
4864:
4837:
4819:
4818:
4800:
4799:
4781:
4780:
4757:
4727:
4707:
4689:
4671:
4651:
4633:
4624:
4606:
4588:
4576:Abstract Algebra
4561:
4560:
4543:
4523:
4517:
4516:
4491:
4485:
4479:
4470:
4464:
4458:
4452:
4446:
4443:
4437:
4436:, p. 24 footnote
4431:
4425:
4419:
4410:
4404:
4398:
4392:
4386:
4380:
4374:
4368:
4359:
4353:
4347:
4341:
4335:
4334:
4332:
4331:
4322:. Archived from
4316:
4310:
4304:
4295:
4289:
4253:
4236:
4230:
4226:
4216:
4210:
4202:Ξ \
4166:
4160:
4144:
4140:
4137:of the subgroup
4136:
4122:
4118: //
4106:
4100:
4094:
4088:
4072:
4056:
4050:
4044:
4028:
4016:
4010:
4004:
3988:
3975:
3969:
3963:
3949:
3945:
3941:
3935:
3931:
3927:
3923:
3919:
3913:
3909:
3905:
3896:
3892:
3888:
3884:
3878:
3869:
3862:
3855:
3849:
3843:
3816:
3810:
3804:
3794:
3788:
3782:
3758:
3748:
3739:
3735:
3727:
3714:
3710:
3691:
3687:
3683:
3673:
3553:
3549:
3545:
3535:
3531:
3523:
3519:
3515:
3511:
3507:
3499:
3495:
3483:
3475:
3471:
3467:
3463:
3459:
3455:
3451:
3429:
3421:
3418:with respect to
3417:
3409:
3398:
3394:
3390:
3386:
3379:
3322:
3318:
3314:
3306:
3291:
3273:
3258:
3240:
3236:
3228:
3224:
3206:
3202:
3200:
3199:
3194:
3192:
3188:
3175:
3161:
3160:
3107:
3103:
3099:
3095:
3093:
3092:
3087:
3085:
3078:
3074:
3073:
3059:
3058:
3016:
3006:
3002:
3001:
2987:
2986:
2938:
2928:
2924:
2923:
2909:
2908:
2873:
2872:
2830:
2821:
2820:
2774:
2770:
2768:
2767:
2762:
2757:
2753:
2752:
2738:
2737:
2684:
2680:
2676:
2674:
2673:
2668:
2663:
2659:
2646:
2626:
2625:
2572:
2560:
2556:
2552:
2545:
2535:
2529:
2525:
2519:
2515:
2511:
2507:
2489:affine subspaces
2486:
2484:
2483:
2478:
2467:
2459:
2451:
2443:
2429:
2414:
2408:
2402:
2396:
2361:
2353:
2340:
2334:
2325:
2321:
2316:congruence class
2313:
2298:
2271:cosets, because
2270:
2266:
2216:
2202:
2192:
2183:
2179:
2173:
2167:
2163:
2146:, +) = ({..., β2
2136:
2130:
2123:
2114:
2110:
2106:
2096:
2082:
2078:
2072:
2040:
2032:
2024:
2017:
2011:
2005:
1997:
1991:
1980:
1961:
1959:
1958:
1953:
1948:
1940:
1917:
1909:
1897:
1891:
1877:
1867:
1861:
1848:
1828:
1813:
1809:
1799:
1790:
1780:
1770:
1762:
1758:
1752:
1746:
1740:
1734:
1724:
1720:
1714:
1710:
1700:
1691:
1678:
1672:
1666:
1662:
1652:
1646:
1642:
1632:
1626:
1617:Normal subgroups
1612:
1594:
1591:with respect to
1590:
1578:
1575:with respect to
1574:
1568:
1562:
1556:
1552:
1546:
1542:
1536:
1530:
1508:
1480:
1472:
1468:
1459:
1449:
1445:
1441:
1437:
1426:
1420:
1414:
1405:
1395:
1391:
1385:
1362:
1352:
1342:
1332:
1322:
1316:
1304:
1284:
1264:
1241:
1224:
1199:
1186:
1177:
1173:
1166:
1162:
1158:
1124:
1114:
1110:
1091:
1081:
1077:
1062:be the subgroup
1061:
1054:
1046:
1021:
993:
971:
964:
945:
937:
914:
897:
875:
871:
860:be the subgroup
859:
850:
845:
840:
833:
828:
823:
813:
801:
794:
789:
784:
774:
769:
764:
757:
752:
745:
740:
735:
725:
720:
713:
708:
703:
693:
688:
683:
676:
667:
662:
652:
647:
642:
635:
630:
623:
613:
608:
603:
596:
591:
586:
579:
569:
564:
559:
552:
547:
539:
531:
518:
504:
469:
420:, respectively.
419:
409:
391:
384:
380:
376:
355:
344:
340:
336:
315:
311:
307:
303:
299:
295:
287:
283:
279:
275:
247:
243:
239:
235:
221:
217:
209:
205:
201:
197:
193:
189:
161:
153:
146:
123:
115:
108:
101:
94:
90:
86:
78:
76:
75:
70:
68:
60:
55:
43:
21:
4893:
4892:
4888:
4887:
4886:
4884:
4883:
4882:
4868:
4867:
4855:
4822:
4804:
4803:
4785:
4784:
4768:
4767:
4764:
4755:
4737:
4734:
4732:Further reading
4725:
4710:
4705:
4692:
4687:
4674:
4669:
4654:
4649:
4639:Basic Algebra I
4636:
4627:
4622:
4609:
4604:
4591:
4586:
4573:
4570:
4565:
4564:
4525:
4524:
4520:
4514:
4493:
4492:
4488:
4480:
4473:
4465:
4461:
4453:
4449:
4444:
4440:
4432:
4428:
4420:
4413:
4405:
4401:
4393:
4389:
4381:
4377:
4369:
4362:
4354:
4350:
4342:
4338:
4329:
4327:
4318:
4317:
4313:
4305:
4298:
4290:
4283:
4278:
4261:
4249: /
4245:
4232:
4228:
4222:
4212:
4206: /
4201:
4196:In geometry, a
4162:
4156:
4152:
4142:
4138:
4132: /
4128: \
4124:
4114:
4102:
4096:
4090:
4074:
4068: /
4064: \
4060:
4052:
4046:
4030:
4024: \
4020:
4012:
4006:
3990:
3984: /
3980:
3971:
3965:
3959:
3956:
3947:
3943:
3937:
3933:
3929:
3925:
3921:
3915:
3911:
3907:
3901:
3900:A double coset
3894:
3890:
3886:
3880:
3874:
3864:
3857:
3851:
3845:
3818:
3812:
3806:
3800:
3790:
3784:
3778:
3775:
3769:
3754:
3741:
3737:
3731:
3716:
3712:
3697:
3689:
3685:
3681:
3671:
3547:
3540:
3533:
3529:
3521:
3517:
3513:
3509:
3505:
3497:
3493:
3481:
3473:
3469:
3465:
3461:
3457:
3453:
3449:
3446:
3440:
3427:
3419:
3415:
3407:
3396:
3392:
3388:
3384:
3377:
3351:conjugate group
3329:
3320:
3316:
3312:
3293:
3279:
3260:
3246:
3238:
3234:
3226:
3222:
3219:
3213:
3204:
3155:
3154:
3149:
3143:
3142:
3137:
3127:
3125:
3121:
3110:
3109:
3105:
3101:
3097:
3083:
3082:
3053:
3052:
3047:
3041:
3040:
3035:
3025:
3023:
3019:
3014:
3008:
3007:
2981:
2980:
2975:
2963:
2962:
2957:
2947:
2945:
2941:
2936:
2930:
2929:
2903:
2902:
2897:
2891:
2890:
2885:
2875:
2867:
2866:
2861:
2855:
2854:
2849:
2839:
2837:
2833:
2828:
2815:
2814:
2809:
2803:
2802:
2797:
2787:
2777:
2776:
2772:
2732:
2731:
2726:
2720:
2719:
2714:
2704:
2702:
2698:
2687:
2686:
2682:
2678:
2620:
2619:
2614:
2608:
2607:
2602:
2592:
2590:
2586:
2575:
2574:
2570:
2567:
2558:
2554:
2547:
2537:
2531:
2527:
2521:
2517:
2513:
2509:
2503:
2417:
2416:
2410:
2404:
2398:
2392:
2381:vector addition
2369:
2357:
2346: /
2342:
2336:
2327:
2326:. The subgroup
2323:
2319:
2300:
2272:
2268:
2218:
2204:
2194:
2185:
2181:
2175:
2169:
2165:
2138:
2132:
2125:
2119:
2112:
2108:
2098:
2088:
2080:
2074:
2042:
2034:
2026:
2019:
2013:
2007:
1999:
1993:
1986:
1976:
1973:
1968:
1900:
1899:
1893:
1887:
1882:, written as .
1873:
1863:
1857:
1844:
1841:
1835:
1815:
1811:
1805: /
1801:
1795:
1782:
1776:
1768:
1760:
1754:
1748:
1742:
1736:
1726:
1722:
1716:
1712:
1706:
1696:
1687: /
1683:
1674:
1668:
1664:
1654:
1648:
1644:
1638:
1635:normal subgroup
1628:
1622:
1619:
1596:
1592:
1583:
1576:
1570:
1564:
1558:
1557:and a subgroup
1554:
1548:
1544:
1538:
1532:
1514:
1504:
1478:
1470:
1461:
1451:
1447:
1443:
1439:
1435:
1422:
1416:
1410:
1397:
1393:
1387:
1364:
1354:
1344:
1334:
1324:
1318:
1314:
1311:
1299:
1293:
1287:
1280:
1273:
1267:
1260:
1250:
1244:
1240:
1233:
1227:
1220:
1210:
1204:
1194:
1188:
1185:
1179:
1175:
1171:
1164:
1160:
1126:
1116:
1112:
1093:
1083:
1079:
1063:
1059:
1052:
1025:
997:
976:
969:
948:
943:
918:
901:
880:
873:
861:
857:
848:
843:
836:
831:
826:
816:
806:
797:
792:
787:
777:
772:
767:
762:
755:
748:
743:
738:
728:
723:
718:
711:
706:
696:
691:
686:
679:
672:
665:
655:
650:
645:
638:
633:
628:
616:
611:
606:
599:
594:
589:
584:
572:
567:
562:
555:
550:
545:
520:
506:
475:
467:
464:
411:
401:
389:
386:
382:
378:
359:
353:
346:
342:
338:
319:
313:
309:
305:
301:
297:
293:
285:
281:
277:
273:
270:
250:normal subgroup
245:
241:
237:
233:
219:
215:
207:
203:
199:
195:
191:
190:. Furthermore,
187:
159:
151:
144:
136:, specifically
121:
110:
103:
96:
92:
88:
84:
46:
45:
41:
35:
28:
23:
22:
15:
12:
11:
5:
4891:
4889:
4881:
4880:
4870:
4869:
4866:
4865:
4853:
4848:
4838:
4820:
4801:
4782:
4769:Nicolas Bray.
4763:
4762:External links
4760:
4759:
4758:
4753:
4733:
4730:
4729:
4728:
4723:
4708:
4703:
4690:
4685:
4672:
4667:
4652:
4647:
4634:
4625:
4620:
4607:
4602:
4589:
4584:
4569:
4566:
4563:
4562:
4518:
4512:
4486:
4471:
4459:
4447:
4438:
4426:
4411:
4409:, pp. 128, 135
4399:
4387:
4375:
4360:
4348:
4336:
4311:
4296:
4280:
4279:
4277:
4274:
4273:
4272:
4267:
4260:
4257:
4256:
4255:
4194:
4183:
4176:
4151:
4148:
4147:
4146:
4112:
4058:
4018:
3955:
3952:
3936:) of the form
3924:an element of
3914:) of the form
3870:respectively.
3838:an element of
3830:an element of
3771:Main article:
3768:
3765:
3749:is called the
3729:if and only if
3665:
3664:
3661:
3658:
3655:
3651:
3650:
3647:
3644:
3641:
3637:
3636:
3633:
3630:
3627:
3623:
3622:
3619:
3616:
3613:
3609:
3608:
3605:
3602:
3599:
3595:
3594:
3591:
3588:
3585:
3581:
3580:
3577:
3574:
3571:
3567:
3566:
3563:
3560:
3557:
3502:standard array
3444:Standard array
3442:Main article:
3439:
3436:
3401:Camille Jordan
3380:of a group of
3328:
3325:
3215:Main article:
3212:
3209:
3191:
3187:
3184:
3181:
3178:
3174:
3170:
3167:
3164:
3159:
3153:
3150:
3148:
3145:
3144:
3141:
3138:
3136:
3133:
3132:
3130:
3124:
3120:
3117:
3081:
3077:
3072:
3068:
3065:
3062:
3057:
3051:
3048:
3046:
3043:
3042:
3039:
3036:
3034:
3031:
3030:
3028:
3022:
3015:
3013:
3010:
3009:
3005:
3000:
2996:
2993:
2990:
2985:
2979:
2976:
2974:
2971:
2968:
2965:
2964:
2961:
2958:
2956:
2953:
2952:
2950:
2944:
2937:
2935:
2932:
2931:
2927:
2922:
2918:
2915:
2912:
2907:
2901:
2898:
2896:
2893:
2892:
2889:
2886:
2884:
2881:
2880:
2878:
2871:
2865:
2862:
2860:
2857:
2856:
2853:
2850:
2848:
2845:
2844:
2842:
2836:
2829:
2827:
2824:
2819:
2813:
2810:
2808:
2805:
2804:
2801:
2798:
2796:
2793:
2792:
2790:
2785:
2784:
2760:
2756:
2751:
2747:
2744:
2741:
2736:
2730:
2727:
2725:
2722:
2721:
2718:
2715:
2713:
2710:
2709:
2707:
2701:
2697:
2694:
2666:
2662:
2658:
2655:
2652:
2649:
2645:
2641:
2638:
2635:
2632:
2629:
2624:
2618:
2615:
2613:
2610:
2609:
2606:
2603:
2601:
2598:
2597:
2595:
2589:
2585:
2582:
2566:
2563:
2476:
2473:
2470:
2466:
2462:
2458:
2454:
2450:
2446:
2442:
2438:
2435:
2432:
2428:
2424:
2368:
2365:
2131:, and now let
1983:additive group
1972:
1969:
1967:
1964:
1951:
1947:
1943:
1939:
1935:
1932:
1929:
1926:
1923:
1920:
1916:
1912:
1908:
1837:Main article:
1834:
1831:
1618:
1615:
1491:representative
1310:
1307:
1306:
1305:
1297:
1291:
1285:
1278:
1271:
1265:
1258:
1248:
1242:
1238:
1231:
1225:
1218:
1208:
1192:
1183:
1049:
1048:
1023:
995:
940:
939:
916:
899:
852:
851:
846:
841:
834:
829:
824:
814:
803:
802:
795:
790:
785:
775:
770:
765:
759:
758:
753:
746:
741:
736:
726:
721:
715:
714:
709:
704:
694:
689:
684:
677:
669:
668:
663:
653:
648:
643:
636:
631:
625:
624:
614:
609:
604:
597:
592:
587:
581:
580:
570:
565:
560:
553:
548:
543:
463:
460:
445:quotient group
439:an element of
371:an element of
358:
331:an element of
318:
269:
266:
118:additive group
81:integers mod 8
67:
63:
59:
54:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4890:
4879:
4876:
4875:
4873:
4862:
4858:
4854:
4852:
4849:
4846:
4842:
4839:
4836:
4832:
4831:
4826:
4821:
4816:
4815:
4810:
4809:"Right Coset"
4807:
4802:
4797:
4796:
4791:
4788:
4783:
4778:
4777:
4772:
4766:
4765:
4761:
4756:
4754:0-486-40922-8
4750:
4746:
4745:
4740:
4736:
4735:
4731:
4726:
4724:0-486-65377-3
4720:
4716:
4715:
4709:
4706:
4700:
4696:
4691:
4688:
4686:9781458500700
4682:
4678:
4673:
4670:
4668:81-224-0120-1
4664:
4660:
4659:
4653:
4650:
4644:
4640:
4635:
4631:
4626:
4623:
4617:
4613:
4608:
4605:
4603:0-06-041601-7
4599:
4595:
4590:
4587:
4585:0-697-06761-0
4581:
4577:
4572:
4571:
4567:
4559:
4555:
4551:
4547:
4542:
4537:
4533:
4529:
4522:
4519:
4515:
4509:
4505:
4501:
4497:
4490:
4487:
4483:
4478:
4476:
4472:
4468:
4463:
4460:
4456:
4451:
4448:
4442:
4439:
4435:
4430:
4427:
4423:
4422:Jacobson 2009
4418:
4416:
4412:
4408:
4403:
4400:
4396:
4391:
4388:
4384:
4379:
4376:
4372:
4371:Fraleigh 1994
4367:
4365:
4361:
4357:
4356:Fraleigh 1994
4352:
4349:
4345:
4340:
4337:
4326:on 2022-01-22
4325:
4321:
4320:"AATA Cosets"
4315:
4312:
4308:
4303:
4301:
4297:
4293:
4288:
4286:
4282:
4275:
4271:
4268:
4266:
4263:
4262:
4258:
4252:
4248:
4244:
4240:
4235:
4225:
4220:
4215:
4209:
4205:
4199:
4195:
4192:
4188:
4184:
4181:
4177:
4174:
4170:
4165:
4159:
4154:
4153:
4149:
4135:
4131:
4127:
4121:
4117:
4113:
4110:
4105:
4099:
4093:
4086:
4082:
4078:
4071:
4067:
4063:
4059:
4055:
4049:
4042:
4038:
4034:
4027:
4023:
4019:
4015:
4009:
4002:
3998:
3994:
3987:
3983:
3979:
3978:
3977:
3974:
3968:
3962:
3953:
3951:
3940:
3918:
3904:
3898:
3883:
3877:
3871:
3867:
3860:
3854:
3848:
3841:
3837:
3833:
3829:
3825:
3821:
3815:
3809:
3803:
3798:
3797:double cosets
3793:
3787:
3781:
3774:
3767:Double cosets
3766:
3764:
3762:
3757:
3752:
3747:
3744:
3740:. The vector
3734:
3730:
3726:
3722:
3719:
3709:
3705:
3701:
3695:
3679:
3675:
3662:
3659:
3656:
3653:
3652:
3648:
3645:
3642:
3639:
3638:
3634:
3631:
3628:
3625:
3624:
3620:
3617:
3614:
3611:
3610:
3606:
3603:
3600:
3597:
3596:
3592:
3589:
3586:
3583:
3582:
3578:
3575:
3572:
3569:
3568:
3564:
3561:
3558:
3555:
3554:
3551:
3543:
3537:
3527:
3503:
3491:
3487:
3486:received word
3479:
3445:
3437:
3435:
3433:
3425:
3413:
3404:
3402:
3383:
3375:
3371:
3366:
3364:
3360:
3356:
3352:
3348:
3344:
3340:
3339:
3334:
3326:
3324:
3310:
3305:
3301:
3297:
3290:
3286:
3282:
3277:
3272:
3268:
3264:
3257:
3253:
3249:
3244:
3232:
3218:
3210:
3208:
3189:
3182:
3176:
3168:
3165:
3162:
3157:
3151:
3146:
3139:
3134:
3128:
3122:
3118:
3115:
3104:is normal in
3079:
3075:
3066:
3063:
3060:
3055:
3049:
3044:
3037:
3032:
3026:
3020:
3011:
3003:
2994:
2991:
2988:
2983:
2977:
2972:
2969:
2966:
2959:
2954:
2948:
2942:
2933:
2925:
2916:
2913:
2910:
2905:
2899:
2894:
2887:
2882:
2876:
2869:
2863:
2858:
2851:
2846:
2840:
2834:
2825:
2822:
2817:
2811:
2806:
2799:
2794:
2788:
2758:
2754:
2745:
2742:
2739:
2734:
2728:
2723:
2716:
2711:
2705:
2699:
2695:
2692:
2664:
2660:
2656:
2653:
2650:
2647:
2639:
2636:
2633:
2630:
2627:
2622:
2616:
2611:
2604:
2599:
2593:
2587:
2583:
2580:
2564:
2562:
2550:
2544:
2540:
2534:
2524:
2506:
2502:
2498:
2494:
2490:
2471:
2468:
2460:
2452:
2444:
2436:
2433:
2430:
2413:
2407:
2401:
2397:, a subspace
2395:
2390:
2386:
2382:
2378:
2377:abelian group
2374:
2373:vector spaces
2366:
2364:
2362:
2360:
2356:integers mod
2354:the group of
2352:
2349:
2345:
2339:
2335:is normal in
2333:
2330:
2317:
2311:
2307:
2304:
2297:
2294:
2290:
2286:
2282:
2278:
2275:
2264:
2260:
2256:
2252:
2248:
2244:
2240:
2236:
2232:
2228:
2224:
2221:
2214:
2210:
2207:
2200:
2197:
2191:
2188:
2178:
2172:
2161:
2157:
2153:
2149:
2145:
2142:
2137:the subgroup
2135:
2128:
2122:
2116:
2105:
2101:
2095:
2091:
2086:
2083:. Due to the
2077:
2070:
2066:
2062:
2058:
2054:
2051:= {..., β6 +
2050:
2046:
2038:
2030:
2023:
2016:
2010:
2003:
1998:the subgroup
1996:
1989:
1984:
1979:
1970:
1966:More examples
1965:
1963:
1949:
1941:
1930:
1927:
1924:
1918:
1910:
1896:
1890:
1885:
1881:
1876:
1871:
1866:
1860:
1856:
1852:
1847:
1840:
1832:
1830:
1827:
1823:
1819:
1808:
1804:
1798:
1792:
1789:
1785:
1779:
1774:
1766:
1757:
1751:
1745:
1739:
1733:
1729:
1719:
1709:
1704:
1699:
1693:
1690:
1686:
1682:
1677:
1671:
1661:
1657:
1651:
1641:
1636:
1631:
1625:
1616:
1614:
1610:
1607:
1603:
1599:
1589:
1586:
1582:
1573:
1567:
1561:
1551:
1541:
1535:
1529:
1525:
1521:
1517:
1512:
1511:abelian group
1507:
1501:
1498:
1496:
1492:
1488:
1484:
1476:
1467:
1464:
1458:
1454:
1433:
1428:
1425:
1419:
1413:
1407:
1404:
1400:
1390:
1383:
1379:
1375:
1371:
1367:
1361:
1357:
1351:
1347:
1341:
1337:
1331:
1327:
1321:
1308:
1303:
1296:
1290:
1286:
1283:
1277:
1270:
1266:
1263:
1257:
1253:
1247:
1243:
1237:
1230:
1226:
1223:
1217:
1213:
1207:
1203:
1202:
1201:
1198:
1191:
1182:
1168:
1156:
1152:
1148:
1144:
1141:
1137:
1133:
1129:
1123:
1119:
1108:
1104:
1100:
1096:
1090:
1086:
1075:
1071:
1067:
1056:
1044:
1040:
1036:
1032:
1028:
1024:
1019:
1016:
1012:
1008:
1004:
1000:
996:
991:
987:
983:
979:
975:
974:
973:
966:
963:
959:
955:
951:
935:
932:
928:
924:
921:
917:
912:
908:
904:
900:
895:
891:
887:
883:
879:
878:
877:
869:
865:
847:
842:
839:
835:
830:
825:
822:
819:
815:
812:
809:
805:
804:
800:
796:
791:
786:
783:
780:
776:
771:
766:
761:
760:
754:
751:
747:
742:
737:
734:
731:
727:
722:
717:
716:
710:
705:
702:
699:
695:
690:
685:
682:
678:
675:
671:
670:
664:
661:
658:
654:
649:
644:
641:
637:
632:
627:
626:
622:
619:
615:
610:
605:
602:
598:
593:
588:
583:
582:
578:
575:
571:
566:
561:
558:
554:
549:
544:
541:
540:
537:
535:
530:
527:
523:
517:
513:
509:
502:
499:
495:
491:
487:
483:
479:
473:
462:First example
461:
459:
458:
454:
450:
446:
442:
438:
434:
430:
426:
421:
418:
414:
408:
404:
399:
394:
374:
370:
366:
362:
357:
351:
334:
330:
326:
322:
317:
291:
267:
265:
263:
259:
258:vector spaces
255:
251:
232:
228:
223:
213:
185:
181:
177:
173:
169:
166:, equal-size
165:
157:
150:
143:
139:
135:
127:
119:
114:
107:
100:
82:
61:
57:
44:is the group
39:
33:
19:
4878:Group theory
4860:
4828:
4812:
4793:
4790:"Left Coset"
4774:
4743:
4714:Group Theory
4713:
4694:
4676:
4657:
4638:
4629:
4611:
4593:
4575:
4541:math/0305256
4531:
4527:
4521:
4495:
4489:
4462:
4450:
4441:
4429:
4402:
4390:
4378:
4351:
4339:
4328:. Retrieved
4324:the original
4314:
4250:
4246:
4237:) that acts
4233:
4223:
4213:
4207:
4203:
4191:Rubik's Cube
4189:for solving
4171:, a type of
4163:
4157:
4133:
4129:
4125:
4119:
4115:
4108:
4103:
4097:
4091:
4084:
4080:
4076:
4069:
4065:
4061:
4053:
4047:
4040:
4036:
4032:
4025:
4021:
4013:
4007:
4000:
3996:
3992:
3985:
3981:
3972:
3966:
3960:
3957:
3938:
3916:
3902:
3899:
3881:
3875:
3872:
3865:
3858:
3852:
3846:
3839:
3835:
3831:
3827:
3823:
3819:
3813:
3807:
3801:
3796:
3791:
3785:
3779:
3776:
3773:Double coset
3755:
3750:
3745:
3742:
3732:
3724:
3720:
3717:
3707:
3703:
3699:
3676:
3668:
3541:
3538:
3526:coset leader
3489:
3485:
3480:(element of
3477:
3447:
3431:
3411:
3405:
3382:permutations
3367:
3358:
3350:
3343:Nebengruppen
3342:
3336:
3330:
3303:
3299:
3295:
3288:
3284:
3280:
3275:
3270:
3266:
3262:
3255:
3251:
3247:
3243:right action
3242:
3220:
3217:Group action
2568:
2553:parallel to
2548:
2542:
2538:
2532:
2522:
2504:
2411:
2405:
2399:
2393:
2370:
2358:
2350:
2347:
2343:
2337:
2331:
2328:
2309:
2305:
2302:
2299:. The coset
2295:
2292:
2288:
2284:
2280:
2276:
2273:
2262:
2258:
2254:
2250:
2246:
2242:
2238:
2234:
2230:
2226:
2222:
2219:
2212:
2208:
2205:
2198:
2195:
2189:
2186:
2176:
2170:
2159:
2155:
2151:
2147:
2143:
2140:
2133:
2126:
2120:
2117:
2103:
2099:
2093:
2089:
2087:of addition
2075:
2068:
2064:
2060:
2056:
2052:
2048:
2044:
2036:
2028:
2021:
2014:
2008:
2001:
1994:
1987:
1977:
1974:
1898:are finite:
1894:
1888:
1879:
1874:
1869:
1864:
1858:
1845:
1842:
1825:
1821:
1817:
1806:
1802:
1796:
1793:
1787:
1783:
1777:
1764:
1755:
1749:
1743:
1737:
1731:
1727:
1717:
1707:
1697:
1694:
1688:
1684:
1675:
1669:
1659:
1655:
1649:
1639:
1629:
1623:
1620:
1608:
1605:
1601:
1597:
1587:
1584:
1571:
1565:
1559:
1549:
1539:
1533:
1527:
1523:
1519:
1515:
1505:
1502:
1499:
1486:
1465:
1462:
1456:
1452:
1429:
1423:
1417:
1411:
1408:
1402:
1398:
1388:
1381:
1377:
1373:
1369:
1365:
1359:
1355:
1349:
1345:
1339:
1335:
1329:
1325:
1319:
1312:
1301:
1294:
1288:
1281:
1275:
1268:
1261:
1255:
1251:
1245:
1235:
1228:
1221:
1215:
1211:
1205:
1196:
1189:
1180:
1169:
1154:
1150:
1146:
1142:
1139:
1135:
1131:
1127:
1121:
1117:
1106:
1102:
1098:
1094:
1088:
1084:
1073:
1069:
1065:
1057:
1050:
1042:
1038:
1034:
1030:
1026:
1017:
1014:
1010:
1006:
1002:
998:
989:
985:
981:
977:
967:
961:
957:
953:
949:
941:
933:
930:
926:
922:
919:
910:
906:
902:
893:
889:
885:
881:
867:
863:
855:
837:
820:
817:
810:
807:
798:
781:
778:
749:
732:
729:
700:
697:
680:
673:
659:
656:
639:
620:
617:
600:
576:
573:
556:
534:Cayley table
528:
525:
521:
515:
511:
507:
500:
497:
493:
489:
485:
481:
477:
465:
456:
455:subgroup of
452:
448:
440:
436:
432:
428:
424:
422:
416:
412:
406:
402:
395:
387:
372:
368:
364:
360:
350:right cosets
349:
347:
332:
328:
324:
320:
289:
271:
231:finite group
224:
180:right cosets
179:
175:
174:. There are
171:
138:group theory
131:
112:
105:
98:
4484:, pp. 14β15
4455:Rotman 2006
4434:Miller 2012
4407:Burton 1988
4395:Rotman 2006
4344:Rotman 2006
4292:Rotman 2006
4169:Vitali sets
3432:right coset
3426:would call
3424:Hall (1959)
3395:instead of
3276:left action
3225:of a group
3221:A subgroup
2487:are called
2415:, the sets
2085:commutivity
1851:cardinality
1814:defined by
1627:of a group
1621:A subgroup
1595:, that is,
1563:of a group
1495:transversal
1469:belongs to
1317:belongs to
423:The symbol
290:left cosets
184:cardinality
176:left cosets
134:mathematics
18:Right coset
4861:groupprops
4845:PlanetMath
4568:References
4467:Scott 1987
4383:Joshi 1989
4330:2020-12-09
4155:Cosets of
3412:left coset
2546:is a line
2229:= {..., β2
2162:, ...}, +)
2102:+ 2 = 2 +
2092:+ 1 = 1 +
1771:is in the
1725:satisfies
1353:such that
1309:Properties
268:Definition
186:) as does
4835:EMS Press
4814:MathWorld
4795:MathWorld
4776:MathWorld
4482:Hall 1959
4307:Dean 1990
3761:linearity
3759:, and by
3500:) into a
3292:given by
3259:given by
3177:−
3169:∈
3163::
3067:∈
3061::
2995:∈
2989::
2917:∈
2911::
2746:∈
2740::
2654:≠
2640:∈
2628::
2493:geometric
2469:∈
2437:∣
2431:∈
2389:subgroups
2385:subspaces
1483:partition
1333:. For if
4872:Category
4558:17839580
4457:, p. 423
4397:, p. 155
4385:, p. 323
4373:, p. 169
4358:, p. 117
4309:, p. 100
4294:, p. 156
4259:See also
4211:, where
4180:transfer
4079: :
4035: :
3995: :
3954:Notation
3826: :
3751:syndrome
3490:decoding
3478:codeword
3355:Burnside
2565:Matrices
2497:parallel
2217:, where
2180:are the
2164:, where
2041:, where
1971:Integers
1855:infinite
435: :
367: :
327: :
164:disjoint
142:subgroup
95:itself,
4857:"Coset"
4771:"Coset"
4469:, p. 19
4424:, p. 52
4346:, p.155
4241:on the
3942:, with
3920:, with
3711:matrix
3327:History
2516:, then
2367:Vectors
2322:modulo
2314:is the
2291:+ 1) =
2203:, ...,
2055:, β3 +
1981:be the
1781:, then
1701:is not
1513:, then
1473:). The
1363:. Thus
470:be the
398:abelian
312:(where
170:called
168:subsets
32:Cosette
4751:
4721:
4701:
4683:
4665:
4645:
4618:
4600:
4582:
4556:
4510:
3795:, the
3736:is in
3696:is an
3688:in an
3663:01010
3649:00011
3635:11010
3621:11001
3607:11111
3593:10011
3579:01011
3565:11011
3456:of an
3349:) and
3333:Galois
3307:. The
3231:action
3017:
2939:
2831:
2530:is in
2383:. The
2379:under
2265:, ...}
2071:, ...}
2067:, 6 +
2063:, 3 +
2033:, and
1773:center
1703:normal
1509:is an
1421:, and
453:normal
288:, the
172:cosets
109:, and
79:, the
4841:Coset
4554:S2CID
4536:arXiv
4276:Notes
4217:is a
3856:when
3660:00111
3657:11100
3654:10001
3646:01110
3643:10101
3640:11000
3632:10111
3629:01100
3626:00001
3618:10100
3615:01111
3612:00010
3604:10010
3601:01001
3598:00100
3590:11110
3587:00101
3584:01000
3576:00110
3573:11101
3570:10000
3562:10110
3559:01101
3556:00000
3468:. As
3466:GF(2)
3347:Weber
3309:orbit
3274:or a
2526:. If
2508:. If
2501:plane
2184:sets
2154:, 0,
1870:index
1862:) as
1820:) β (
1633:is a
1489:is a
1323:then
1145:} = {
1037:} = {
1022:, and
1009:} = {
972:are:
915:, and
876:are:
451:is a
212:index
162:into
149:group
147:of a
128:is 4.
126:index
4749:ISBN
4719:ISBN
4699:ISBN
4681:ISBN
4663:ISBN
4643:ISBN
4616:ISBN
4598:ISBN
4580:ISBN
4508:ISBN
4265:Heap
4095:and
3970:and
3958:Let
3932:(in
3910:(in
3889:and
3879:and
3863:and
3805:and
3783:and
3706:) Γ
3410:the
3372:was
3302:) β
3269:) β
2569:Let
2312:, +)
2215:β 1)
2097:and
1992:and
1975:Let
1892:and
1824:) =
1765:Some
1442:and
1170:Let
1125:and
1115:are
1092:and
1082:are
1058:Let
960:} =
856:Let
519:and
466:Let
377:for
348:The
337:for
272:Let
260:and
178:and
140:, a
111:3 +
104:2 +
97:1 +
4843:at
4546:doi
4532:273
4500:doi
4161:in
4141:in
4101:in
4089:of
4083:in
4077:KgH
4051:in
4045:of
4039:in
4011:in
4005:of
3999:in
3946:in
3939:hxK
3917:Hxk
3903:HxK
3882:HyK
3876:HxK
3868:= 1
3861:= 1
3824:hgk
3822:= {
3820:HgK
3811:in
3799:of
3753:of
3414:of
3311:of
3237:on
3233:of
2681:of
2409:in
2318:of
2257:, 2
2237:, β
2211:+ (
2201:+ 1
2174:in
2158:, 2
2150:, β
2039:+ 2
2031:+ 1
2012:in
1878:in
1872:of
1826:abN
1791:.)
1775:of
1715:in
1705:in
1695:If
1673:in
1647:of
1637:of
1547:of
1537:of
1503:If
1450:if
1368:= (
1130:= {
1097:= {
1029:= {
1001:= {
984:= {
952:= {
950:abT
925:= {
905:= {
888:= {
410:or
388:As
381:in
363:= {
341:in
323:= {
308:of
296:in
292:of
284:of
240:of
218:in
214:of
206:in
198:in
158:of
156:set
132:In
4874::
4859:.
4833:,
4827:,
4811:.
4792:.
4773:.
4552:,
4544:,
4530:,
4506:,
4474:^
4414:^
4363:^
4299:^
4284:^
4221:,
4033:Hg
3993:gH
3950:.
3897:.
3834:,
3723:=
3702:β
3430:a
3428:gH
3408:gH
3399:.
3323:.
3321:Hg
3317:gH
3304:hg
3298:,
3287:β
3283:Γ
3278:,
3271:gh
3265:,
3254:β
3250:Γ
3245:,
3207:.
2685:,
2561:.
2541:+
2363:.
2308:+
2283:=
2279:+
2261:+
2253:+
2249:,
2245:,
2241:+
2233:+
2225:+
2193:,
2059:,
2047:+
2025:,
2000:(3
1822:bN
1818:aN
1788:Ha
1786:=
1784:aH
1732:Hb
1730:=
1728:aH
1692:.
1660:Ng
1658:=
1656:gN
1613:.
1609:Hg
1600:=
1598:Hg
1588:Hg
1526:+
1522:=
1518:+
1457:yH
1455:=
1453:xH
1438:,
1406:.
1401:=
1399:aH
1382:aH
1376:=
1370:ga
1366:xH
1358:=
1356:ga
1348:β
1340:gH
1338:β
1330:xH
1328:=
1326:gH
1320:gH
1300:β
1274:β
1254:β
1236:Hg
1234:=
1229:Hg
1214:=
1195:β
1187:,
1167:.
1155:ba
1153:,
1151:ba
1149:,
1138:,
1136:ab
1134:,
1128:Hb
1120:=
1118:HI
1107:ba
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