# O'Nan group

In the area of modern 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'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/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.

O'N is one of the 6 sporadic simple groups called the pariahs because (Griess 1982) showed that it is not a subquotient of the monster group.

## 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:

• 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)

## References

• 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
• R. L. Griess, Jr, The Friendly Giant, Inventiones Mathematicae 69 (1982), 1-102. p. 94: proof that O'N is a pariah.
• A. J. E. Ryba, A new construction of the O'Nan simple group. J. Algebra 112 (1988), no. 1, 173-197.MR 0921973
• 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 783183