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 normal closure of a subset of a group
Definition
Let be a group and let . The family
is nonempty because by Normal subgroup: invariance under conjugation. Its intersection is normal by The intersection of a nonempty family of normal subgroups is normal. The normal closure of in is
It contains and is contained in every normal subgroup of that contains . Thus it is the smallest normal subgroup of containing .
Depends on
Used by
- If one set in a van Kampen cover is simply connected, the other fundamental group surjects with overlap-generated kernel Corollary
- An HNN extension with its stable letter Definition
- Group presentation by generators and relations Definition
- Tietze transformations: dictionary generators, redundant relators, renaming, and their inverses Definition
- A relator set and its symmetrisation have the same normal closure Lemma
- Reidemeister-Schreier relators are independent of word representatives Lemma
- The boundary label of a van Kampen diagram is trivial in the presented group Lemma
- Each Tietze transformation preserves the isomorphism type of the presented group Proposition
- The normal closure of R is the set of finite products of conjugates of elements of R and their inverses Proposition
- A group pushout is the quotient of a free product by the amalgamating relations Theorem
- Every group admits a presentation Theorem
- Grp is complete and cocomplete Theorem
- Von Dyck's theorem: maps of generators that satisfy the relators extend uniquely from a presented group Theorem
Dependency tree · two levels
6 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
- Encyclopedia of Mathematics, HNN-extension (standard reference, not scraped)