CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 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.

A transitive action is faithful exactly when a point stabiliser is core-free

Statement

Let $G$ act transitively on a nonempty set $X$ , and let $x \in X$ . The action is faithful if and only if

${Core}_{G} (G_{x}) = {e} .$

Equivalently, faithful transitive $G$ -sets are precisely the coset actions $G / H$ for core-free subgroups $H$ .

Facts & Assumptions

Given: A transitive action of $G$ on a nonempty set $X$ and a point $x \in X$ .

[L1]

An action is faithful when the only group element fixing every point is the identity (Left group actions, transitive actions, and faithful actions).

[L2]

The action on $X$ is equivariantly isomorphic to the left-coset action on $G / G_{x}$ (Every transitive $G$ -set is equivariantly isomorphic to $G / G_{x}$ for any chosen point $x$ ).

[L3]

The kernel of the action on $G / H$ is ${Core}_{G} (H)$ (Left multiplication on $G / H$ is transitive, has stabiliser $H$ at $H$ , and has kernel ${Core}_{G} (H)$ ).

[L4]

The core is the intersection of all conjugates of the subgroup (The core ${Core}_{G} (H) = ⋂_{g \in G} g H g^{- 1}$ of a subgroup).

Proof

technique · direct

1.1

By [L2], the equivariant isomorphism identifies the given action with the action on $G / G_{x}$ .

2.1

A group element fixes every point of $X$ exactly when it fixes every point of the equivariantly isomorphic coset set; by [L3] and [L4], the set of such elements is ${Core}_{G} (G_{x})$ .

step 1.1L3L4

3.1

By [L1], the action is faithful exactly when this kernel is ${e}$ , proving both directions.

step 2.1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

15 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)