LemmaStatement: Literature-sourcedProof: 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.

k-transitivity implies k-homogeneity and lower transitivity

Statement

Let $1 \leq j \leq k$ , and let $G$ act on a set $Ω$ with at least $k$ distinct points. If the action is $k$ -transitive, then it is $k$ -homogeneous and also $j$ -transitive.

Facts & Assumptions

Given: Integers $1 \leq j \leq k$ , a $G$ -action on a set $Ω$ with at least $k$ distinct points, and the action is $k$ -transitive.

[L1]

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

Proof

technique · direct

1.1L1choose

To prove $k$ -homogeneity, let $A, B \subseteq Ω$ be $k$ -element subsets. Choose orderings $A = {α_{1}, \dots, α_{k}}$ and $B = {β_{1}, \dots, β_{k}}$ . By [L1], some $g \in G$ sends each $α_{i}$ to $β_{i}$ , so $g \cdot A = B$ .

1.2L1choose

To prove $j$ -transitivity, start with ordered $j$ -tuples of distinct points $(α_{1}, \dots, α_{j})$ and $(β_{1}, \dots, β_{j})$ . Because $Ω$ has at least $k$ distinct points, extend them to ordered $k$ -tuples of distinct points $(α_{1}, \dots, α_{k})$ and $(β_{1}, \dots, β_{k})$ . Then [L1] gives $g \in G$ with $g \cdot α_{i} = β_{i}$ for all $1 \leq i \leq k$ , in particular for $1 \leq i \leq j$ .

2.1step 1.1step 1.2∎

Step 1.1 gives $k$ -homogeneity and step 1.2 gives $j$ -transitivity.

Depends on

k-transitive and k-homogeneous actions

Used by

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

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)