DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge 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-transitive and k-homogeneous actions

Definition

Let $k \geq 1$ , and let $G$ act on a set $Ω$ .

The action is $k$ -transitive if for any ordered $k$ -tuples $(α_{1}, \dots, α_{k}), (β_{1}, \dots, β_{k})$ of pairwise distinct points of $Ω$ , there is some $g \in G$ with $g \cdot α_{i} = β_{i} for every 1 \leq i \leq k .$

The action is $k$ -homogeneous if for any $k$ -element subsets $A, B \subseteq Ω$ , there is some $g \in G$ with $g \cdot A = B$ .

Depends on

Left group actions, transitive actions, and faithful actions

Used by

A transitive action on more than one point is doubly transitive exactly when it has rank two Corollary
The square-affine group of F₇ is 2-homogeneous but not 2-transitive Counterexample
Sharply k-transitive actions Definition
Affine general linear groups are doubly transitive Example
The natural actions of symmetric and alternating groups Example
k-transitivity implies k-homogeneity and lower transitivity Lemma
Every doubly transitive action is primitive Proposition

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)