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.
The orbit and stabilizer of a point in a group action
Definition
For a left action of on and , the orbit of is
and the stabilizer of is
The subgroup claim implicit in the word “stabilizer” is proved in The stabilizer is a subgroup 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
- Rank, suborbits, and subdegrees of a transitive action Definition
- The cycle construction CYC(A) 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 two-circle wedge has both regular and nonregular connected three-sheeted coverings Example
- If gcd(‖ a‖,m)=1 then the shift stabiliser of a is trivial, so its orbit has exactly m elements Lemma
- If y=g· x, then G_y=gGₓg⁻¹ Lemma
- The stabilizer Gₓ is a subgroup of G Lemma
- Every permutation of a finite set is a product of pairwise disjoint cycles, uniquely up to reordering and cyclic rotation Theorem
- Orbit-stabiliser: G/Gₓ→ G· x, gGₓ↦ g· x, is a well-defined bijection 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 · 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
- Brosnan, Orbits and stabilizers (standard reference, not scraped)