O'Nan group

In the mathematical field of group theory, the O'Nan group or O'Nan–Sims group O'N is a sporadic simple group of order

   2⁹ · 3⁴ · 5 · 7³ · 11 · 19 · 31

= 460815505920

found by Michael O'Nan (1976).

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 they are not found within the Monster group.

History

The O'Nan group 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).PSL_{3(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.}

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 classes of maximal subgroups of the O'Nan group given 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
• 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 

• Atlas of Finite Group Representations: O'Nan group