O'Nan–Scott theorem
In mathematics, the O'Nan–Scott theorem is one of the most influential theorems of permutation group theory; the classification of finite simple groups is what makes it so useful. Originally the theorem was about maximal subgroups of the symmetric group. It appeared as an appendix to a paper by Leonard Scott written for The Santa Cruz Conference on Finite Groups in 1979, with a footnote that Michael O'Nan had independently proved the same result.
The theorem states that a maximal subgroup of the symmetric group Sym(Ω), where |Ω| = n is one of the following:
In their paper, "On the O'Nan Scott Theorem for primitive permutation groups," M.W. Liebeck, Cheryl Praeger and Jan Saxl give a complete self-contained proof of the theorem. In addition to the proof, they recognized that real power in the O'Nan–Scott theorem is in the ability to split the finite primitive groups into various types.
The eight O'Nan–Scott types are as follows: HA (holomorph of an abelian group): These are the primitive groups which are subgroups of the affine general linear group AGL(d,p), for some prime p and positive integer d ≥ 1. For such a group G to be primitive, it must contain the subgroup of all translations, and the stabilizer G0 in G of the zero vector must be an irreducible subgroup of GL(d,p). Primitive groups of type HA are characterized by having a unique minimal normal subgroup which is elementary abelian and acts regularly.
HS (holomorph of a simple group): Let T be a finite nonabelian simple group. Then M = T×T acts on Ω = T by t(t1,t2) = t1−1tt2. Now M has two minimal normal subgroups N1, N2, each isomorphic to T and each acts regularly on Ω, one by right multiplication and one by left multiplication. The action of M is primitive and if we take α = 1T we have Mα = {(t,t)|t ∈ T}, which includes Inn(T) on Ω. In fact any automorphism of T will act on Ω. A primitive group of type HS is then any group G such that M ≅ T.Inn(T) ≤ G ≤ T.Aut(T). All such groups have N1 and N2 as minimal normal subgroups.
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