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.

Primitive and imprimitive transitive actions

Definition

Let $G$ act transitively on a set $Ω$ (Left group actions, transitive actions, and faithful actions).

The action is primitive if every block $B \subseteq Ω$ is trivial: either $B = Ω$ or $B = {ω}$ for some $ω \in Ω$ .

The action is imprimitive if it is transitive and not primitive, that is, if it admits a block that is neither a singleton nor all of $Ω$ .

When $Ω$ is nonempty, this is equivalently the condition that the only $G$ -invariant block systems are the singleton partition ${{ω} : ω \in Ω}$ and the one-block partition ${Ω}$ . The block formulation also covers the empty transitive action without treating ${\emptyset}$ as a partition into nonempty blocks.

Depends on

Used by

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)