O'Nan–Scott theorem
= O'Nan–Scott theorem
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The theorem classifies finite primitive permutation groups by their minimal normal subgroups and the action of their socle. The modern types are affine (HA), almost simple (AS), simple diagonal (SD), compound diagonal (CD), product action (PA), holomorph simple (HS), holomorph compound (HC), and twisted wreath (TW). A coarser version describes primitive maximal subgroups of symmetric groups by affine, almost-simple, diagonal and full product-action overgroups. Intransitive and imprimitive maxima are respectively unequal-part subset stabilizers and block-system stabilizers. Primitivity alone is not maximality in the symmetric group; parity can put an otherwise natural candidate entirely inside the alternating group.