O'Nan group
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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 · 5 · 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.
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'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
- ^ Griess (1982): p. 94: proof that O'N is a pariah
- 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 (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, 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