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

Statement

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

Facts & Assumptions

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

[L1]

In the quotient group, $(g N) (h N) = g h N$ (For $N ⊴ G$ , the cosets form a group with identity $N$ and inverse $(g N)^{- 1} = g^{- 1} N$ ).

[F1]

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

Proof

technique · direct

1.1

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

L1F1

2.1

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

step 1.1F1∎

Depends on

Used by

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

T. W. Judson, Abstract Algebra: Theory and Applications, Normal Subgroups and Factor Groups, Exercises (standard reference, not scraped)