DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 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

Dependency tree · two levels

10 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

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