In the area of abstract algebra known as group theory, the O'Nan group O'N or O'NanâÂÂSims group is a sporadic simple group of order
O'N is one of the 26 sporadic groups and was found by 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/2<sup>n</sup>Z ÃÂZ/2<sup>n</sup>Z ÃÂZ/2<sup>n</sup>Z).PSL<sub>3</sub>(F<sub>2</sub>). In O'Nan's original paper, the definition of Alperin type required that n âÂÂ¥ 2, because the n = 1 case had already been classified. If this restriction is removed, then the following simple groups have Sylow 2-subgroups of Alperin type:
The Schur multiplier has order 3, and its outer automorphism group has order 2. showed that O'N cannot be a subquotient of the monster group. Thus it is one of the 6 sporadic groups called the pariahs.
showed that its triple cover has two 45-dimensional representations over the field with 7 elements, exchanged by an outer automorphism.
The degrees of irreducible representations of the O'Nan group are 1, 10944, 13376, 13376, 25916, ... .
and independently found the 13 conjugacy classes of maximal subgroups of O'N as follows:
In 2017 John F. R. Duncan, Michael H. Mertens, and Ken Ono proved theorems that establish an analogue of monstrous moonshine for the O'Nan group. Their results "reveal a role for the O'Nan pariah group as a provider of hidden symmetry to quadratic forms and elliptic curves." The O'Nan moonshine results "also represent the intersection of moonshine theory with the Langlands program, which, since its inception in the 1960s, has become a driving force for research in number theory, geometry and mathematical physics." .
An informal description of these developments was written by in Quanta Magazine.