ExampleConstruction: AI-adaptedVerification: 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.

$G / {e}$ reproduces $G$ , while $G / G$ is the one-element quotient group

Example

For every group $G$ with identity $e$ , the quotient $G / {e}$ consists of the singleton cosets ${g}$ and has exactly the same multiplication as $G$ after identifying $g {e}$ with $g$ . At the other extreme, $G / G = {G}$ is the one-element quotient group.

Facts & Assumptions

Given: A group $G$ with identity $e$ .

[F1]

The sets ${e}$ and $G$ are subgroups of $G$ (Subgroup).

[F2]

A left coset is $g N = {g n : n \in N}$ (Left and right cosets $g H$ and $H g$ of a subgroup).

[L1]

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

Verification

technique · direct

1.1

For every $g \in G$ , [F2] gives $g {e} = {g e} = {g}$ . Hence the cosets of ${e}$ are precisely the singleton subsets of $G$ .

1.2

The subgroup ${e}$ is normal because $g {e} g^{- 1} = {e}$ , and [L1] gives $(g {e}) (h {e}) = (g h) {e}$ . Thus $g {e} \mapsto g$ preserves the multiplication exactly.

F1L1algebra

2.1

For every $g \in G$ , [F2] gives $g G = G$ , so $G / G$ has the sole element $G$ . Since $G$ is normal in itself, [L1] makes this the one-element quotient group.

F1F2L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

Encyclopedia of Mathematics, Quotient group (standard reference, not scraped)

G/{e}G/{e} reproduces GG, while G/GG/G is the one-element quotient group