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 object of an abelian category has a composition series
Statement
Every object of an abelian category has a composition series.
Facts & Assumptions
Given: The abelian group .
A composition series is a finite strict chain with simple successive quotients (Composition series and composition factors of an object).
Finite-length objects are exactly those admitting composition series (Object of finite length).
Refutation
Every nonzero subgroup of is of the form , hence is isomorphic to . So if a composition series existed, the first nonzero term would satisfy , and the first quotient would not be simple. This contradicts [L1].
Therefore has no composition series, so by [L2] it is not of finite length. The universal statement is false even in .
Depends on
Used by
Nothing in the library uses this result yet.
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
- Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, and Victor Ostrik, Tensor Categories, Section 1.5 (standard reference, not scraped)