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.
FALSE: every HNN extension is an ascending HNN extension
Statement
Every HNN extension is the ascending HNN extension of some injective endomorphism of its base group.
Facts & Assumptions
Given: The definitions of a general HNN extension and of an ascending HNN extension.
An ascending HNN extension has one associated subgroup equal to the whole base group. (Ascending HNN extensions of injective endomorphisms)
A general HNN extension only requires two injectively embedded associated subgroups. (An HNN extension with its stable letter)
Refutation
Let and choose associated subgroups and with the isomorphism . By [L2] this defines an HNN extension.
Neither associated subgroup is all of , so this example cannot satisfy the condition in [L1]. Therefore it is an HNN extension that is not ascending, and the statement is false.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
9 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
- C. Loh, Geometric Group Theory: An Introduction (2015 course version) (standard reference, not scraped)