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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
Contents
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. 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: