CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-26

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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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A transitive action of prime degree is primitive

Statement

Let $G$ act transitively on a finite set $Ω$ of prime cardinality. Then the action is primitive.

Facts & Assumptions

Given: A transitive action of $G$ on a finite set $Ω$ with $∣ Ω ∣$ prime.

[L1]

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

[L2]

In a finite transitive action, every block in a block system has the same size, and that size divides $∣ Ω ∣$ (Blocks in a finite transitive action have a common size).

[L3]

A prime natural number has no positive divisors other than $1$ and itself (Prime and composite integers: $p$ is prime when $p > 1$ and its only positive divisors are $1$ and $p$ ).

Proof

technique · direct

1.1L2L3

Let $B$ be a block system and let $B \in B$ . By [L2], the positive integer $∣ B ∣$ divides the prime $∣ Ω ∣$ . So [L3] gives $∣ B ∣ = 1$ or $∣ B ∣ = ∣ Ω ∣$ .

2.1step 1.1L1L2∎

If $∣ B ∣ = 1$ , then every block has size $1$ by [L2], so $B$ is the singleton partition. If $∣ B ∣ = ∣ Ω ∣$ , then $B = Ω$ and $B = {Ω}$ . Thus every block system is trivial, and [L1] makes the action primitive.

Depends on

Used by

Dihedral actions of prime and composite degree Example

Dependency tree · two levels

19 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)