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 fixed-point sets and of a group action
Definition
Let a group act on a set (Left group actions, transitive actions, and faithful actions). For , the fixed-point set of is
The global fixed-point set is
Depends on
Used by
- A finite p-group action on X has a global fixed point whenever p∤|X| Corollary
- S₃ acting on three points has |X|=3 and |X^S₃|=0, so the fixed-point congruence modulo 2 fails without the p-group hypothesis Counterexample
- Colourings, weight functions, and the pattern inventory Definition
- Fixed subtrees and minimal invariant subtrees Definition
- The cycle index of a finite permutation group Definition
- The cycle-index series of a graded family of Sₙ-actions Definition
- Cauchy-Frobenius orbit counting: |G| |X/G|=∑_g∈ G|Xᵍ| for a finite group action Theorem
- If a finite p-group P acts on a finite set X, then |X|≡|X^P| (mod p) 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
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, 14.3 (standard reference, not scraped)