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. Michael Aschbacher and Scott later gave a corrected version of the statement of the theorem.

The theorem states that a maximal subgroup of the symmetric group Sym(Ω), where |Ω| = n, is one of the following:

  1. Sk × Sn−k the stabilizer of a k-set (that is, intransitive)
  2. Sa wr Sb with n = ab, the stabilizer of a partition into b parts of size a (that is, imprimitive)
  3. primitive (that is, preserves no nontrivial partition) and of one of the following types:
  • AGL(d,p)
  • Sl wr Sk, the stabilizer of the product structure Ω = Δk
  • a group of diagonal type
  • an almost simple group

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. A complete version of the theorem with a self-contained proof was given by M.W. Liebeck, Cheryl Praeger and Jan Saxl. The theorem is now a standard part of textbooks on permutation groups.

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