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O'Nan group

In the mathematical field of group theory, the O'Nan group O'N (found by Michael O'Nan (1976)) is a sporadic simple group of order

   2⁹ · 3⁴ · 5 · 7³ · 11 · 19 · 31

= 460815505920

≈ 5 · 10¹¹.

The Schur multiplier has order 3, and its outer automorphism group has order 2. Its triple cover has two 45-dimensional representations over the field with 7 elements, exchanged by an outer automorphism (Ryba 1988). The subgroup fixed by an outer automorphism of order 2 is the Janko group J₁, a maximal subgroup.

O'N is one of the 6 sporadic simple groups called the pariahs, because they are not found within the Monster group. As J₁ is a subgroup, O'N must be a pariah, but J₁ was not identified as a pariah until 1986 and O'N's pariah status was proven directly before then.

References

• M. E. O'Nan, Some evidence for the existence of a new simple group, Proc. London Math Soc. 32 (1976) 421-479.
• R. L. Griess, Jr, The Friendly Giant, Inventiones Mathematicae 69 (1982), 1-102. p. 94: proof that O'N is a pariah.
• Robert A. Wilson, Is J₁ a subgroup of the monster?, Bull. London Math. Soc. 18, no. 4 (1986), 349-350.
• A. J. E. Ryba, A new construction of the O'Nan simple group. J. Algebra 112 (1988), no. 1, 173-197.MR0921973
• MathWorld: O'Nan Group
• Atlas of Finite Group Representations: O'Nan group