ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck 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.

The natural actions of symmetric and alternating groups

Example

For $n \geq 2$ , the natural action of $S_{n}$ on ${1, \dots, n}$ is sharply $n$ -transitive. For $n \geq 4$ , the natural action of $A_{n}$ on the same set is $(n - 2)$ -transitive.

Facts & Assumptions

Given: The natural permutation actions of $S_{n}$ and $A_{n}$ on ${1, \dots, n}$ .

[L1]

For $k \geq 1$ , a sharply $k$ -transitive action has a unique group element carrying any ordered $k$ -tuple of distinct points to any other such tuple (Sharply k-transitive actions).

[L2]

For $k \geq 1$ , a $k$ -transitive action carries any ordered $k$ -tuple of distinct points to any other such tuple (k-transitive and k-homogeneous actions).

Verification

technique · direct

1.1L1

For the action of $S_{n}$ , a permutation is determined uniquely by the images of the ordered tuple $(1, \dots, n)$ , and every ordered $n$ -tuple of distinct points is another listing of ${1, \dots, n}$ . So this action is sharply $n$ -transitive by [L1].

1.2L2choose

For the action of $A_{n}$ with $n \geq 4$ , take two ordered $(n - 2)$ -tuples of distinct points and let $c_{1}, c_{2}$ and $d_{1}, d_{2}$ be the two complementary points. Some $σ \in S_{n}$ sends the first full $n$ -tuple $(α_{1}, \dots, α_{n - 2}, c_{1}, c_{2})$ to the second $(β_{1}, \dots, β_{n - 2}, d_{1}, d_{2})$ .

2.1step 1.2L2∎

If $σ \in A_{n}$ , then $σ$ already sends $α_{i}$ to $β_{i}$ for $1 \leq i \leq n - 2$ . If $σ \notin A_{n}$ , compose it with the transposition $(d_{1} d_{2})$ , which fixes each $β_{i}$ and reverses parity. So in either case there is an even permutation sending the first $(n - 2)$ -tuple to the second. Hence the natural action of $A_{n}$ is $(n - 2)$ -transitive.

Depends on

Used by

FALSE: every nontrivial normal subgroup of a faithful primitive group is regular False statement

Dependency tree · two levels

3 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)