DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-11

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.

Equivariant maps and isomorphisms of group actions

Definition

Let $X$ and $Y$ be $G$ -sets (Left group actions, transitive actions, and faithful actions). A function $f:X\to Y$ (A function is a relation $f$ with $(a,b) \in f$ and $(a,c) \in f$ implying $b = c$ ; $f : A \to B$ , the value $f(a)$ , domain and codomain) is $G$ -equivariant when

$f(g\cdot x)=g\cdot f(x)$

for every $g\in G$ and $x\in X$ . An isomorphism of $G$ -sets is an equivariant bijection (Injection, surjection, bijection). Two actions are equivariantly isomorphic when such a bijection exists between their $G$ -sets.

Depends on

Used by

Every transitive G-set is equivariantly isomorphic to G/Gₓ for any chosen point x Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 26 results over 15 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

P. Brosnan, Undergraduate Algebra Notes, 3.14: G-Sets (standard reference, not scraped)