ExampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02

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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 $gN=\{gn:n\in N\}$ (Left and right cosets $gH$ and $Hg$ of a subgroup).

[L1]

For a normal subgroup $N$ , quotient multiplication is $(gN)(hN)=(gh)N$ (For $N\mathrel{\trianglelefteq}G$ , the cosets form a group with identity $N$ and inverse $(gN)^{-1}=g^{-1}N$ ).

Verification

technique · direct

1.1

For every $g\in G$ , [F2] gives $g\{e\}=\{ge\}=\{g\}$ . 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 $(g\{e\})(h\{e\})=(gh)\{e\}$ . Thus $g\{e\}\mapsto g$ preserves the multiplication exactly.

F1L1algebra

2.1

For every $g\in G$ , [F2] gives $gG=G$ , 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

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Sources

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

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