False statementConstruction: Literature-sourcedVerification: AI-generatedprecheck passverified 2026-09-24 (gpt-6-sol)

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FALSE: the cited LPS O'Nan-Scott proof uses no CFSG consequence

Statement

False claim: The cited Liebeck–Praeger–Saxl (LPS) proof of the five-type O'Nan–Scott classification uses no consequence of the classification of finite simple groups (CFSG).

Facts & Assumptions

Given: The proof in Liebeck–Praeger–Saxl, On the O'Nan–Scott Theorem for Finite Primitive Permutation Groups (1988), pp. 389–396.

[F1]

In Case 2(a), on printed p. 394, LPS let $Y$ be the kernel of the action on the simple direct factors. They say that $Y_{α}$ embeds in a product of outer automorphism groups and is therefore soluble "by the Schreier 'Conjecture'". They then use solubility in the commutator argument proving $Y = M$ .

[F2]

On printed p. 395, in the simple-socle case with trivial socle point stabilizer, LPS again say that $G_{α}$ is soluble "by the Schreier 'Conjecture'". They use this to choose a minimal normal subgroup $Q$ of $G_{α}$ which is elementary abelian, and continue the exclusion argument on printed pp. 395–396.

[F3]

The Schreier theorem says that the outer automorphism group of every finite nonabelian simple group is soluble. Smith, Applying the Classification of Finite Simple Groups, Theorem 1.5.1 (printed p. 23), identifies it as a consequence of CFSG.

Refutation

technique · direct

1.1F1F2

The cited LPS proof invokes Schreier explicitly in two branches. In Case 2(a), the invoked solubility is used to prove $Y = M$ ; in the simple-socle branch, it supplies the elementary-abelian minimal normal subgroup used in the subsequent argument. These are proof steps, not merely remarks about later refinements.

2.1F3step 1.1∎

Schreier is a CFSG consequence. Thus this specific five-type proof does use a CFSG consequence, refuting the stated claim. This establishes neither that CFSG is logically necessary for the theorem nor that another proof could not avoid it.

Used by

Nothing in the library uses this result yet.

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Liebeck, Praeger and Saxl, On the O'Nan-Scott Theorem for Finite Primitive Permutation Groups, J. Austral. Math. Soc. Ser. A 44 (1988), 389–396 (standard reference, not scraped)
Stephen D. Smith, Applying the Classification of Finite Simple Groups: A User's Guide, §1.5 and §6.1 (standard reference, not scraped)