DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-03

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.

Left group actions, transitive actions, and faithful actions

Definition

Let $G$ be a group with identity $e$ and let $X$ be a set. A left action of $G$ on $X$ is a function $G \times X \to X$ , written $(g, x) \mapsto g \cdot x$ , such that

$e \cdot x = x, (g h) \cdot x = g \cdot (h \cdot x)$

for all $g, h \in G$ and $x \in X$ . Then $X$ is a $G$ -set. The action is transitive when every $x, y \in X$ satisfy $g \cdot x = y$ for some $g \in G$ . It is faithful when $g \cdot x = x$ for every $x \in X$ implies $g = e$ .

Left group actions, transitive actions, and faithful actions

Definition

Depends on

Dependency tree · 2 levels

Sources