The O'Nan-Scott classification of finite primitive groups (recorded)

Every finite primitive permutation group of degree at least $2$ belongs to exactly one of the five coarse O'Nan-Scott types used on this page: affine, almost simple, diagonal, product action, or twisted wreath.

Liebeck–Praeger–Saxl (LPS), printed p. 392, states the five-branch theorem in the convention used here. Its proof on pp. 392–396 is a genuine case analysis, not a consequence merely of the socle split or of the five definitions. The previous local [A2] assumed exactly the theorem to be proved.

In Case 2(a), printed p. 394, LPS uses solvability of outer automorphism groups of finite simple groups (the Schreier theorem) to prove the kernel $Y$ of the factor action equals the socle $M$; this is needed to identify the twisted-wreath action. At the end, printed pp. 395–396, it uses Schreier again to exclude a regular simple socle in the almost-simple branch and to complete the product-action branch. The complete local proof of these steps, or an alternative exact five-type proof, is still missing. This item therefore records the cited classification without claiming it is proved here.