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 $g = e$ .

Depends on

Group and abelian group

Used by

A transitive action is faithful exactly when a point stabiliser is core-free Corollary
Every Riemann surface is a quotient of a simply connected model Corollary
G-sets are strictly monadic over sets Corollary
Mackey little-group reduction for an abelian normal subgroup Corollary
A nontransitive system with two orbits is not classified by one stabilizer Counterexample
A transitive action need not be Frobenius Counterexample
Collisions destroy freeness of the coordinate permutation action Counterexample
The action of ℤ/2 on two disjoint two-point orbits is free but not transitive Counterexample
The natural action of S₃ on three points is faithful and transitive but not free Counterexample
A free group action has no nonidentity element fixing a point Definition
A group acting on a ring by automorphisms and its invariant subring Definition
Affine root hyperplanes, coroot translations, alcoves, and the affine reflection group Definition
An action of a group H on a group N by automorphisms Definition
An R-linear action of G on a left R-module, and a G-module over R Definition
Blocks and block systems for a group action Definition
Colourings, weight functions, and the pattern inventory Definition
Compositions, partial flags, and standard parabolics Definition
Covering-space actions by disjoint translates of neighbourhoods Definition
Crossed homomorphism for a G-group Definition
Cuspidal representations and Harish-Chandra series Definition
Deck transformations and the deck-transformation group of a covering Definition
Equivariant maps and isomorphisms of group actions Definition
Faithful weak action Definition
Finite-group randomization setups Definition
Free and properly discontinuous group actions Definition
Graph automorphisms and group actions on a simplicial graph Definition
Harish-Chandra induction and restriction for finite general linear groups Definition
Isometric, proper, and cobounded actions on metric spaces Definition
k-transitive and k-homogeneous actions Definition
Measurable finite-group permutation setups Definition
Ordered partitions and coordinate parabolics Definition
Permutation groups, stabilizers, supports, and normal filters Definition
Pointwise nonlinearities, neighborhood pooling, subgroup sampling, and coset pooling Definition
Primitive and imprimitive transitive actions Definition
Regular actions Definition
Systems of imprimitivity for a Borel G-space Definition
The cycle construction CYC(A) Definition
The cycle index of a finite permutation group Definition
The finite-holonomy normal model of a compact leaf Definition
The fixed-point sets Xᵍ and X^G of a group action Definition

…and 75 more results.

Dependency tree · two levels

6 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

Brosnan, Group actions (standard reference, not scraped)