PropositionStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-08-26

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Every doubly transitive action is primitive

Statement

Every doubly transitive action is primitive.

Facts & Assumptions

Given: A doubly transitive action of $G$ on $Ω$ .

[L1]

A block $B$ satisfies: for every $g \in G$ , either $g \cdot B = B$ or $(g \cdot B) \cap B = \emptyset$ (Blocks and block systems for a group action).

[L2]

A $2$ -transitive action sends any ordered pair of distinct points to any other such pair (k-transitive and k-homogeneous actions).

[L3]

A transitive action is primitive when its only block systems are the singleton partition and the one-block partition (Primitive and imprimitive transitive actions).

Proof

technique · direct

1.1L1choose

Let $B$ be a block containing some $α \in Ω$ . If $B = {α}$ there is nothing to prove, so suppose $B$ also contains $β \neq α$ .

1.2L1L2

For any $γ \neq α$ , [L2] gives an element $g \in G$ with $g \cdot α = α$ and $g \cdot β = γ$ . Then $γ \in g \cdot B$ , while $α \in g \cdot B \cap B$ because $g$ fixes $α$ . So [L1] gives $g \cdot B = B$ , and therefore $γ \in B$ .

2.1step 1.1step 1.2L3∎

Step 1.2 shows that every $γ \neq α$ lies in $B$ , so $B = Ω$ . Thus any block containing more than one point is all of $Ω$ , and the only block systems are the trivial ones. By [L3], the action is primitive.

Depends on

Used by

Affine general linear groups are doubly transitive Example
Projective linear actions and Iwasawa's hypotheses Example
FALSE: every nontrivial normal subgroup of a faithful primitive group is regular False statement
FALSE: the O'Nan-Scott theorem is the classification of finite simple groups False statement
Finite 2-transitive groups have affine or almost simple socle type Proposition

Dependency tree · two levels

4 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

J. S. Milne, Group Theory, Chapter 4 (standard reference, not scraped)
K. Conrad, Transitive Group Actions (standard reference, not scraped)