CorollaryStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-02

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.

Every quotient group of an abelian group is abelian

If $G$ is abelian and $N ⊴ G$ , then $G / N$ is abelian.

Given: An abelian group $G$ and a normal subgroup $N ⊴ G$ .

[L1]

[F1]

A group is abelian when $g h = h g$ for all of its elements (Group and abelian group).

technique · direct

1.1L1F1

For arbitrary cosets $g N, h N \in G / N$ , commutativity in $G$ gives $(g N) (h N) = g h N = h g N = (h N) (g N)$ .

2.1step 1.1F1∎

Hence every two elements of $G / N$ commute, so $G / N$ is abelian.

Depends on