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

• 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.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 812997 
• 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 

External links

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