PropositionStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-27

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Finite 2-transitive groups have affine or almost simple socle type

Statement

Every finite $2$ -transitive permutation group of degree at least $2$ is of affine type or almost simple type.

Facts & Assumptions

Given: A finite $2$ -transitive permutation group $G \leq Sym (Ω)$ of degree at least $2$ .

[L1]

Every doubly transitive action is primitive (Every doubly transitive action is primitive).

[A1]

In the finite O'Nan-Scott classification, the diagonal, product-action, and twisted-wreath types do not occur for $2$ -transitive groups.

Proof

technique · direct

1.1

By [L1], the $2$ -transitive action of $G$ is primitive, so the O'Nan-Scott classification applies to it.

givenL1

2.1

The source fact [A1] removes the diagonal, product-action, and twisted-wreath branches from the primitive classification. Therefore the only remaining possibilities are affine type and almost simple type.

A1step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

15 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Leonard H. Soicher, Primitive permutation groups (standard reference, not scraped)