TheoremStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck 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.

For $N\mathrel{\trianglelefteq}G$ , the cosets form a group with identity $N$ and inverse $(gN)^{-1}=g^{-1}N$

Statement

Let $N\mathrel{\trianglelefteq}G$ . The left cosets form a group $G/N$ under

$(gN)(hN)=ghN.$

Its identity is $N=eN$ , and the inverse of $gN$ is $g^{-1}N$ .

Facts & Assumptions

Given: A group $G$ and a normal subgroup $N\mathrel{\trianglelefteq}G$ .

[L1]

Coset multiplication $(gN)(hN)=ghN$ is well defined when $N$ is normal (Coset multiplication $(gH)(hH)=ghH$ is well defined if and only if $H$ is normal).

[F1]

The quotient set $G/N$ consists of the left cosets of $N$ , with the proposed product $(gN)(hN)=ghN$ (The quotient group $G/N$ and coset product $(gN)(hN)=ghN$ ).

[F2]

A group operation is associative, has a two-sided identity, and gives every element a two-sided inverse (Group and abelian group).

Proof

technique · direct

1.1

By [L1], the formula in [F1] is a binary operation on the coset set, independent of representatives.

L1F1

1.2

For $g,h,k\in G$ , one has $((gN)(hN))(kN)=(gh)kN=g(hk)N=(gN)((hN)(kN))$ . Also $(eN)(gN)=gN=(gN)(eN)$ , so $N=eN$ is the identity.

F1F2algebra

1.3

The products $(gN)(g^{-1}N)$ and $(g^{-1}N)(gN)$ both equal $eN=N$ , so $g^{-1}N$ is the inverse of $gN$ .

F1F2algebra

2.1

Steps 1.1 through 1.3 verify the binary operation, associativity, identity, and inverse axioms; therefore $G/N$ is a group with the stated identity and inverses.

step 1.1step 1.2step 1.3F2∎

Depends on

Used by

Cited to discharge well-definedness by The quotient group G/N and coset product (gN)(hN)=ghN.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 22 results over 16 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

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

For N⊴GN\mathrel{\trianglelefteq}GN⊴G, the cosets form a group with identity NNN and inverse (gN)−1=g−1N(gN)^{-1}=g^{-1}N(gN)−1=g−1N