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 be a group with identity and let be a set. A left action of on is a function , written , such that
for all and . Then is a -set. The action is transitive when every satisfy for some . It is faithful when for every implies .
Depends on
Used by
- A transitive action is faithful exactly when a point stabiliser is core-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
- A free group action has no nonidentity element fixing a point Definition
- Equivariant maps and isomorphisms of group actions Definition
- The fixed-point sets Xᵍ and X^G of a group action Definition
- The orbit G· x and stabilizer Gₓ of a point in a group action Definition
- An action groupoid has the acted-on set as objects, orbits as connected components, and stabilizers as automorphism groups Example
- 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
- The trivial action of ℤ/2 on a singleton is transitive but not faithful Example
- There are six binary necklaces of length four up to rotation Example
- There are six two-colourings of the vertices of a square up to its eight symmetries Example
- If y=g· x, then G_y=gGₓg⁻¹ Lemma
- The stabilizer Gₓ is a subgroup of G Lemma
- Actions of G on X correspond exactly to homomorphisms GtoSym(X) Theorem
- Cayley's theorem: every group G is isomorphic to a subgroup of Sym(G) Theorem
- Every permutation of a finite set is a product of pairwise disjoint cycles, uniquely up to reordering and cyclic rotation Theorem
- Every transitive G-set is equivariantly isomorphic to G/Gₓ for any chosen point x Theorem
- Jordan's derangement theorem: every transitive action of a finite group on a finite set with more than one element has a nonidentity element with no fixed points Theorem
- Left multiplication on G/H is transitive, has stabiliser H at H, and has kernel Core_G(H) Theorem
- The orbits of a group action are the equivalence classes of x∼ y iff y=g· x for some g, and hence partition the acted-on set Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 9 results over 9 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
- Brosnan, Group actions (standard reference, not scraped)