DefinitionDefinition: AI-adaptedProof: 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.

Rank, suborbits, and subdegrees of a transitive action

Definition

Let $G$ act transitively on a set $Ω$ , and fix a point $α \in Ω$ . Its stabilizer is $G_{α} : = {g \in G : g \cdot α = α}$ (The orbit $G \cdot x$ and stabilizer $G_{x}$ of a point in a group action).

The orbits of $G_{α}$ on $Ω$ are the suborbits of the action at $α$ .

Their cardinalities are the subdegrees.

The rank of the transitive action is the number of its suborbits at the chosen basepoint $α$ .

Depends on

The orbit $G\cdot x$ and stabilizer $G_x$ of a point in a group action

Used by

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)