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.
Blocks and block systems for a group action
Definition
Let act on a set by a left action (Left group actions, transitive actions, and faithful actions), and let for .
A nonempty subset is a block for this action if for every one has either
A block system is a partition of into blocks.
Every singleton is a block, and each orbit of the action is a block. These are the trivial examples to which later definitions compare all other block systems.
Depends on
Used by
- Primitive and imprimitive transitive actions Definition
- Blocks in a regular cyclic action are cosets of subgroups Example
- Dihedral actions of prime and composite degree Example
- The imprimitive wreath product preserves its fiber blocks Example
- The translates of a block partition its orbit Lemma
- Blocks in a finite transitive action have a common size Proposition
- Every doubly transitive action is primitive Proposition
- A transitive imprimitive action embeds modulo its kernel in an imprimitive wreath product Theorem
- Blocks containing a point correspond to intermediate subgroups Theorem
- G-invariant block systems are exactly the invariant equivalence relations 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
- J. S. Milne, Group Theory, Chapter 4 (standard reference, not scraped)
- K. Conrad, Transitive Group Actions (standard reference, not scraped)