DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 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 free group action has no nonidentity element fixing a point
Definition
A left action of a group on a set (Left group actions, transitive actions, and faithful actions) is free when
for every and . Equivalently, no nonidentity element of fixes any point of .
Depends on
Used by
- 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
- Left multiplication gives a free and transitive action of every group on itself Example
- The four rotations of a square act freely, transitively and faithfully on its vertices Example
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 7 results over 7 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- T. W. Judson, Abstract Algebra: Theory and Applications, 14.1 (standard reference, not scraped)