False statementConstruction: Literature-sourcedVerification: AI-adaptedprecheck passverified 2026-09-09 (gpt-6-astra)

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FALSE: the O'Nan-Scott theorem is the classification of finite simple groups

Statement

False claim: the O'Nan-Scott theorem is the classification of finite simple groups.

Facts & Assumptions

Given: The names refer to these two classification questions: O'Nan–Scott asks for the socle and action types of finite primitive permutation groups; the classification of finite simple groups asks for the abstract isomorphism types of all finite simple groups. Comparing the questions does not assume the conclusions or proofs of either classification.

[L1]

A simple group is nontrivial and has no proper nontrivial normal subgroup (Simple groups).

[L2]

The symmetric group consists of all permutations of a set (The symmetric group $Sym (X)$ : the bijections of a set $X$ under composition). A $2$ -transitive action moves any ordered pair of distinct points to any other, and is primitive (k-transitive and k-homogeneous actions, Every doubly transitive action is primitive).

Refutation

technique · direct

1.1givenL2algebra

The natural action of $S_{3}$ on ${1, 2, 3}$ is $2$ -transitive: specifying the images of two distinct points determines a permutation by sending the third point to the remaining point. It is therefore primitive by [L2], and lies in the domain of the O'Nan–Scott classification question.

2.1step 1.1givenL1algebra∎

The subgroup $A = {1, (123), (132)}$ is nontrivial and proper in $S_{3}$ . Conjugating either $3$ -cycle by a permutation merely relabels its three entries, so it gives one of these same two $3$ -cycles. Thus $A$ is normal and $S_{3}$ is not simple by [L1]. Consequently the two classification questions have different domains: one includes this action of a nonsimple group, whereas the other classifies simple groups up to abstract isomorphism. Their conclusions also ask for different data, action types versus a list of abstract simple groups. They are not the same theorem.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

Liebeck, Praeger and Saxl, On the O'Nan-Scott theorem for finite primitive permutation groups, Introduction and Section 2 (standard reference, not scraped)