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.

The fixed-point sets $X^{g}$ and $X^{G}$ of a group action

Definition

Let a group $G$ act on a set $X$ (Left group actions, transitive actions, and faithful actions). For $g \in G$ , the fixed-point set of $g$ is

$X^{g} : = {x \in X : g \cdot x = x} .$

The global fixed-point set is

$X^{G} : = {x \in X : g \cdot x = x for every g \in G} = ⋂_{g \in G} X^{g} .$

Depends on

Left group actions, transitive actions, and faithful actions

Dependency tree · 2 levels

Sources

T. W. Judson, Abstract Algebra: Theory and Applications, 14.3