# O'Nan group

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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
    460,815,505,920 = 2$^9$ · 3$^4$ · 5 · 7$^3$ · 11 · 19 · 31 ≈ 5×10$^{11}$.

## History

O'N 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/2$^n$Z ×Z/2$^n$Z ×Z/2$^n$Z).PSL$_3$(F$_2$). The following simple groups have Sylow 2-subgroups of Alperin type:
* For the Chevalley group G$_2$(q), if q is congruent to 3 or 5 mod 8, n = 1 and the extension does not split.
* For the Steinberg group $^3$D$_4$(q), if q is congruent to 3 or 5 mod 8, n = 1 and the extension does not split.
* For the alternating group A$_8$, 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'N cannot be a subquotient of the monster group. Thus it is one of the 6 sporadic groups called the pariahs.

## Representations

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

Wilson (1985) and Yoshiara (1985) independently found the 13 conjugacy classes of maximal subgroups of O'N as follows:
Maximal subgroups of O'N :
| No. | Structure | Order | Index | Comments |
| 1,2 | L$_3$(7):2 | 3,753,792 = 2$^6$·3$^2$·7$^3$·19 | 122,760 = 2$^3$·3$^2$·5·11·31 | two classes, fused by an outer automorphism |
| 3 | J$_1$ | 175,560 = 2$^3$·3·5·7·11·19 | 2,624,832 = 2$^6$·3$^3$·7$^2$·31 | the subgroup fixed by an outer involution in O'N:2 |
| 4 | 4$_2$$^·$L$_3$(4):2$_1$ | 161,280 = 2$^9$·3$^2$·5·7 | 2,857,239 = 3$^2$·7$^2$·11·19·31 | the centralizer of an (inner) involution in O'N |
| 5 | (3$^2$:4 × A$_6$)$^·$2 | 25,920 = 2$^6$·3$^4$·5 | 17,778,376 = 2$^3$·7$^3$·11·19·31 | |
| 6 | 3$^4$:2$^{1+4}$.D$_{10}$ | 25,920 = 2$^6$·3$^4$·5 | 17,778,376 = 2$^3$·7$^3$·11·19·31 | |
| 7,8 | L$_2$(31) | 14,880 = 2$^5$·3·5·31 | 30,968,784 = 2$^4$·3$^3$·7$^3$·11·19 | two classes, fused by an outer automorphism |
| 9 | 4$^{3·}$L$_3$(2) | 10,752 = 2$^9$·3·7 | 42,858,585 = 3$^3$·5·7$^2$·11·19·31 | |
| 10,11 | M$_{11}$ | 7,920 = 2$^4$·3$^2$·5·11 | 58,183,776 = 2$^5$·3$^2$·7$^3$·19·31 | two classes, fused by an outer automorphism |
| 12,13 | A$_7$ | 2,520 = 2$^3$·3$^2$·5·7 | 182,863,296 = 2$^6$·3$^2$·7$^2$·11·19·31 | two classes, fused by an outer automorphism |

## O'Nan moonshine

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

* 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