# O'Nan group

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

## History

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). For the O'Nan group n = 2 and the extension does not split. The only other simple group with a Sylow 2-subgroup of Alperin type with n ≥ 2 is the Higman–Sims group again with n = 2, but the extension splits.

The Schur multiplier has order 3, and its outer automorphism group has order 2.

In 1982 R. L. Griess showed that O'Nan cannot be a subquotient of the monster group.[1] 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'Nan as follows:

• L3(7):2 (2 classes, fused by an outer automorphism)
• J1 The subgroup fixed by an outer involution.
• 42.L3(4):21
• (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

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."[2] In September 2017 Erica Klarreich of Quanta Magazine published an article [1] explaining these recent developments in an informal way.

## References

1. Griess (1982): p. 94: proof that O'Nan is a pariah
2. 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. PMID 28935903.
• 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
• R. L. Griess, Jr (1982). The Friendly Giant, Inventiones Mathematicae, vol 69, issue 1, doi:10.1007/BF01389186
• A. J. E. Ryba, A new construction of the O'Nan simple group. J. Algebra 112 (1988), no. 1, 173-197.MR0921973
• 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
• 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