Alphabeta MathHow 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
- Groups acting freely without inversions on trees are torsion-free Corollary
- 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
- Covering-space actions by disjoint translates of neighbourhoods Definition
- Regular actions Definition
- 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
- A permutation group with a regular normal subgroup G embeds in Hol(G) Proposition
- On a connected covering space, a deck transformation is determined by one point and the deck action is free Proposition
- Left translation acts on a Cayley graph by label-preserving automorphisms, freely on vertices Theorem
Dependency tree · two levels
2 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
- T. W. Judson, Abstract Algebra: Theory and Applications, 14.1 (standard reference, not scraped)