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.
The external direct product with componentwise multiplication
Definition
Let and be groups. Their external direct product has underlying set
and componentwise operation
The fact that this operation makes a group, with the indicated identity and inverses, is proved in is a group with identity , coordinatewise inverses, and homomorphic coordinate projections ↗. Until that result is used, this definition introduces only the set and its componentwise binary operation.
Depends on
Used by
- Elementary-divisor data for a finite abelian group Definition
- Internal direct products of finitely many normal subgroups Definition
- Invariant-factor data for a finite abelian group Definition
- The canonical surjection from a free product to the direct product of its factors Example
- FALSE: every finite group is a direct product of cyclic prime-power groups False statement
- G× H is a group with identity (e_G,e_H), coordinatewise inverses, and homomorphic coordinate projections Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 9 results over 9 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
- Sharifi, Abstract Algebra, direct products (standard reference, not scraped)