O'Nan group

O'Nan 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). The following simple groups have Sylow 2-subgroups of Alperin type:

• For the Chevalley group G₂(q), if q is congruent to 3 or 5 mod 8, n = 1 and the extension does not split.
• For the Steinberg group ³D₄(q), if q is congruent to 3 or 5 mod 8, n = 1 and the extension does not split.
• For the alternating groupA₈, n = 1 and the extension splits.
• For the O'Nan group, n = 2 and the extension does not split.
• For the Higman-Sims group, n = 2 and the extension splits.

The Schur multiplier has order 3, and its outer automorphism group has order 2. (Griess 1982:94) showed that O'Nan cannot be a subquotient of the monster group. Thus it is one of the 6 sporadic groups called the pariahs.

Ryba (1988) showed that its triple cover has two 45-dimensional representations over the field with 7 elements, exchanged by an outer automorphism.

Wilson (1985) and Yoshiara (1985) independently found the 13 conjugacy classesofmaximal subgroupsofO'Nan as follows:

• L₃(7):2 (2 classes, fused by an outer automorphism)
• J₁ The subgroup fixed by an outer involutioninO'Nan:2.
• 4₂.L₃(4):2₁ The centralizer of an (inner) involutioninO'Nan.
• (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)

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 symmetrytoquadratic 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." (Duncan, Mertens & Ono 2017, article 670).

An informal description of these developments was written by Erica Klarreich (2017) in Quanta Magazine.

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