Alphabeta Math
PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-03
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.

For finite groups G and H, ∣G×H∣=∣G∣ ∣H∣

Statement

If G and H are finite groups, then their external direct product is finite and has order ∣G×H∣=∣G∣ ∣H∣.

Facts & Assumptions

Given: Finite groups G,H.

[L1]

The direct product has underlying set G×H and is a group (G×H is a group with identity (eG,eH), coordinatewise inverses, and homomorphic coordinate projections).

[L2]

The order of a finite group is the cardinality of its underlying set (The order ∣G∣ of a finite group and the order ord⁡(g) of an element, with ord⁡(g)=∞ when no positive power of g is the identity).

[L3]

The Cartesian product of finite sets has cardinality the product of their cardinalities (The product rule: ∣A×B∣=∣A∣ ∣B∣, and ∣∏i<mAi∣=∏i<m∣Ai∣).

Proof

technique · direct
1.1L1given

The carrier of the direct-product group is the Cartesian product of the finite carriers G and H.

2.1step 1.1L3

By the finite product rule, G×H is finite and ∣G×H∣set=∣G∣set∣H∣set.

3.1step 2.1L2∎

Reading these three cardinalities as group orders gives ∣G×H∣=∣G∣ ∣H∣.

Depends on

Dependency tree · 2 levels

Sources