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O'Nan group

January 01, 1970

In the area of abstract algebra known as group theory, the O'Nan group O'N or O'Nan–Sims group is a sporadic simple group of order

   29 · 34 · 5 · 73 · 11 · 19 · 31
= 460815505920
≈ 5×1011.

History edit

O'Nan is one of the 26 sporadic groups and was found by Michael O'Nan (1976) in a study of groups with a Sylow 2-subgroup of "Alperin type", meaning isomorphic to a Sylow 2-Subgroup of a group of type (Z/2nZ ×Z/2nZ ×Z/2nZ).PSL3(F2). The following simple groups have Sylow 2-subgroups of Alperin type:

  • For the Chevalley group G2(q), if q is congruent to 3 or 5 mod 8, n = 1 and the extension does not split.
  • For the Steinberg group 3D4(q), if q is congruent to 3 or 5 mod 8, n = 1 and the extension does not split.
  • For the alternating group A8, n = 1 and the extension splits.
  • For the O'Nan group, n = 2 and the extension does not split.
  • For the Higman-Sims group, n = 2 and the extension splits.

The Schur multiplier has order 3, and its outer automorphism group has order 2. (Griess 1982:94) showed that O'Nan cannot be a subquotient of the monster group. Thus it is one of the 6 sporadic groups called the pariahs.

Representations edit

Ryba (1988) showed that its triple cover has two 45-dimensional representations over the field with 7 elements, exchanged by an outer automorphism.

Maximal subgroups edit

Wilson (1985) and Yoshiara (1985) independently found the 13 conjugacy classes of maximal subgroups of O'Nan as follows:

  • L3(7):2 (2 classes, fused by an outer automorphism)
  • J1 The subgroup fixed by an outer involution in O'Nan:2.
  • 42.L3(4):21 The centralizer of an (inner) involution in O'Nan.
  • (32:4 × A6).2
  • 34:21+4.D10
  • L2(31) (2 classes, fused by an outer automorphism)
  • 43.L3(2)
  • M11 (2 classes, fused by an outer automorphism)
  • A7 (2 classes, fused by an outer automorphism)

O'Nan moonshine edit

In 2017 John F. R. Duncan, Michael H. Mertens, and Ken Ono proved theorems that establish an analogue of monstrous moonshine for the O'Nan group. Their results "reveal a role for the O'Nan pariah group as a provider of hidden symmetry to quadratic forms and elliptic curves." The O'Nan moonshine results "also represent the intersection of moonshine theory with the Langlands program, which, since its inception in the 1960s, has become a driving force for research in number theory, geometry and mathematical physics." (Duncan, Mertens & Ono 2017, article 670).

An informal description of these developments was written by Erica Klarreich (2017) in Quanta Magazine.

Sources edit

  • Duncan, John F. R.; Mertens, Michael H.; Ono, Ken (2017), "Pariah moonshine", Nature Communications, 8 (1), Article number: 670, doi:10.1038/s41467-017-00660-y, PMC 5608900, PMID 28935903
  • Griess, R. L. (1982), "The Friendly Giant", Inventiones Mathematicae, 69 (1): 1007, Bibcode:1982InMat..69....1G, doi:10.1007/BF01389186, hdl:2027.42/46608
  • Klarreich, Erica (22 September 2017). "Moonshine Link Discovered for Pariah Symmetries". Quanta Magazine. Retrieved 23 August 2020.
  • O'Nan, Michael E. (1976), "Some evidence for the existence of a new simple group", Proceedings of the London Mathematical Society, Third Series, 32 (3): 421–479, doi:10.1112/plms/s3-32.3.421, ISSN 0024-6115, MR 0401905
  • Ryba, A. J. E. (1988), "A new construction of the O'Nan simple group", Journal of Algebra, 112 (1): 173–197, doi:10.1016/0021-8693(88)90141-X, MR 0921973
  • Wilson, Robert A. (1985), "The maximal subgroups of the O'Nan group", Journal of Algebra, 97 (2): 467–473, doi:10.1016/0021-8693(85)90059-6, ISSN 0021-8693, MR 0812997
  • Yoshiara, Satoshi (1985), "The maximal subgroups of the sporadic simple group of O'Nan", Journal of the Faculty of Science. University of Tokyo. Section IA. Mathematics, 32 (1): 105–141, ISSN 0040-8980, MR 0783183

External links edit

  • MathWorld: O'Nan Group
  • "Atlas of Finite Group Representations: O'Nan group".

January 01, 1970
group, area, abstract, algebra, known, group, theory, sims, group, sporadic, simple, group, order, 460815505920, 1011, contents, history, representations, maximal, subgroups, moonshine, sources, external, linkshistory, edito, sporadic, groups, found, michael, . In the area of abstract algebra known as group theory the O Nan group O N or O Nan Sims group is a sporadic simple group of order 29 34 5 73 11 19 31 460815505920 5 1011 Contents 1 History 2 Representations 3 Maximal subgroups 4 O Nan moonshine 5 Sources 6 External linksHistory editO Nan is one of the 26 sporadic groups and was found by Michael O Nan 1976 in a study of groups with a Sylow 2 subgroup of Alperin type meaning isomorphic to a Sylow 2 Subgroup of a group of type Z 2nZ Z 2nZ Z 2nZ PSL3 F2 The following simple groups have Sylow 2 subgroups of Alperin type For the Chevalley group G2 q if q is congruent to 3 or 5 mod 8 n 1 and the extension does not split For the Steinberg group 3D4 q if q is congruent to 3 or 5 mod 8 n 1 and the extension does not split For the alternating group A8 n 1 and the extension splits For the O Nan group n 2 and the extension does not split For the Higman Sims group n 2 and the extension splits The Schur multiplier has order 3 and its outer automorphism group has order 2 Griess 1982 94 showed that O Nan cannot be a subquotient of the monster group Thus it is one of the 6 sporadic groups called the pariahs Representations editRyba 1988 showed that its triple cover has two 45 dimensional representations over the field with 7 elements exchanged by an outer automorphism Maximal subgroups editWilson 1985 and Yoshiara 1985 independently found the 13 conjugacy classes of maximal subgroups of O Nan as follows L3 7 2 2 classes fused by an outer automorphism J1 The subgroup fixed by an outer involution in O Nan 2 42 L3 4 21 The centralizer of an inner involution in O Nan 32 4 A6 2 34 21 4 D10 L2 31 2 classes fused by an outer automorphism 43 L3 2 M11 2 classes fused by an outer automorphism A7 2 classes fused by an outer automorphism O Nan moonshine editIn 2017 John F R Duncan Michael H Mertens and Ken Ono proved theorems that establish an analogue of monstrous moonshine for the O Nan group Their results reveal a role for the O Nan pariah group as a provider of hidden symmetry to quadratic forms and elliptic curves The O Nan moonshine results also represent the intersection of moonshine theory with the Langlands program which since its inception in the 1960s has become a driving force for research in number theory geometry and mathematical physics Duncan Mertens amp Ono 2017 article 670 An informal description of these developments was written by Erica Klarreich 2017 in Quanta Magazine Sources editDuncan John F R Mertens Michael H Ono Ken 2017 Pariah moonshine Nature Communications 8 1 Article number 670 doi 10 1038 s41467 017 00660 y PMC 5608900 PMID 28935903 Griess R L 1982 The Friendly Giant Inventiones Mathematicae 69 1 1007 Bibcode 1982InMat 69 1G doi 10 1007 BF01389186 hdl 2027 42 46608 Klarreich Erica 22 September 2017 Moonshine Link Discovered for Pariah Symmetries Quanta Magazine Retrieved 23 August 2020 O Nan Michael E 1976 Some evidence for the existence of a new simple group Proceedings of the London Mathematical Society Third Series 32 3 421 479 doi 10 1112 plms s3 32 3 421 ISSN 0024 6115 MR 0401905 Ryba A J E 1988 A new construction of the O Nan simple group Journal of Algebra 112 1 173 197 doi 10 1016 0021 8693 88 90141 X MR 0921973 Wilson Robert A 1985 The maximal subgroups of the O Nan group Journal of Algebra 97 2 467 473 doi 10 1016 0021 8693 85 90059 6 ISSN 0021 8693 MR 0812997 Yoshiara Satoshi 1985 The maximal subgroups of the sporadic simple group of O Nan Journal of the Faculty of Science University of Tokyo Section IA Mathematics 32 1 105 141 ISSN 0040 8980 MR 0783183External links editMathWorld O Nan Group Atlas of Finite Group Representations O Nan group Retrieved from https en wikipedia org w index php title O 27Nan group amp oldid 1223972786, wikipedia, wiki, book, books, library,

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