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.
[1] Michael Aschbacher and Scott later gave a corrected version of the statement of the theorem.
[2] The theorem states that a maximal subgroup of the symmetric group Sym(Ω), where |Ω| = n, is one of the following: In a survey paper written for the Bulletin of the London Mathematical Society, Peter J. Cameron seems to have been the first to recognize that the real power in the O'Nan–Scott theorem is in the ability to split the finite primitive groups into various types.
Primitive groups of type HA are characterized by having a unique minimal normal subgroup which is elementary abelian and acts regularly.
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.
A group of type HC preserves a product structure Ω = Δk where Δ = T and G≤ HwrSk where H is a primitive group of type HS on Δ. TW (twisted wreath): Here G has a unique minimal normal subgroup N and N ≅ Tk for some finite nonabelian simple group T and N acts regularly on Ω.
The group Sk induces automorphisms of N by permuting the entries and fixes the subgroup H and so acts on the set Ω.
Moreover, N = Tk ◅ G and G induces a transitive subgroup of Sk in its action on the k simple direct factors of N. Some authors use different divisions of the types.