O'Nan group

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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[edit]

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).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.

Representations[edit]

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[edit]

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[edit]