PropositionStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-26

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A finite sharply k-transitive action has order n(n-1)...(n-k+1)

Statement

Let $G$ act sharply $k$ -transitively on a finite set $Ω$ of size $n$ , with $k \leq n$ . Then $∣ G ∣ = n (n - 1) \dots (n - k + 1) .$

Facts & Assumptions

Given: A sharply $k$ -transitive action of $G$ on a finite set $Ω$ of size $n$ , with $k \leq n$ .

[L1]

In a sharply $k$ -transitive action, for any two ordered $k$ -tuples of distinct points there is a unique group element carrying the first tuple to the second (Sharply k-transitive actions).

Proof

technique · direct

1.1

Fix one ordered $k$ -tuple of distinct points $(α_{1}, \dots, α_{k})$ . Define $Φ : G \to T_{k} (Ω)$ by $Φ (g) : = (g \cdot α_{1}, \dots, g \cdot α_{k})$ , where $T_{k} (Ω)$ is the set of ordered $k$ -tuples of distinct points of $Ω$ .

L1construct

2.1

The map $Φ$ is bijective: existence in [L1] makes it surjective, and uniqueness in [L1] makes it injective.

L1step 1.1

3.1

The set $T_{k} (Ω)$ has $n$ choices for the first entry, then $n - 1$ for the second, and so on down to $n - k + 1$ for the last. Hence $∣ T_{k} (Ω) ∣ = n (n - 1) \dots (n - k + 1)$ , and step 2.1 gives the same value for $∣ G ∣$ .

step 2.1algebra∎

Depends on

Sharply k-transitive actions

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 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)