How 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.
Internal direct products of finitely many normal subgroups
Definition
Let be a group and let be normal subgroups, where . They form an internal direct product when they generate and, for each , The empty family is an internal direct product of the trivial group. For two subgroups of an abelian group this says and ; in additive notation one writes . Normal subgroups and generated subgroups are those of Normal subgroup: invariance under conjugation and The subgroup generated by a subset, the cyclic subgroup , and cyclic groups, and the comparison product is The external direct product with componentwise multiplication.
Depends on
Used by
- Indecomposable and decomposable nontrivial finite abelian groups Definition
- Complements of a maximal cyclic subgroup in Cₚ times Cₚ need not be unique Example
- The unit group modulo one hundred is isomorphic to C₂0 times C₂ Example
- A maximal-order cyclic subgroup splits off a finite abelian p-group Theorem
- Internal direct products are external direct products, equivalently every element has a unique factorisation Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 13 results over 10 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- Keith Conrad, Decomposition of Finite Abelian Groups, §§1-4 (standard reference, not scraped)
- Richard Elman, Lectures on Abstract Algebra, Ch. 14 (standard reference, not scraped)