Alphabeta MathCyclic Groups and Direct Products
About this page
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.
✓ 5 results · all verified · 0 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 5 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Abstract Algebra
Number Theory
Real Analysis
2 · Logical flowchart
3 · Items
Definition
The external direct product G× H with componentwise multiplication
Theorem
G× H is a group with identity (e_G,e_H), coordinatewise inverses, and homomorphic coordinate projections
Proposition
For finite groups G and H, |G× H|=|G| |H|
Theorem
If g and h have finite orders m and n, then ι(ord(g,h))=lcm(ι(m),ι(n)) in G× H
Theorem
Every cyclic group is isomorphic to (ℤ,+) or to (ℤ/n,+) for its finite order n≥1
Theorem