CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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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.
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A set-theoretic section of C_4 onto C_2 need not be a homomorphism

Statement refuted

Every set-theoretic section of a quotient map of groups is automatically a homomorphism.

The quotient map $C_{4} \to C_{2}$ with section $[0] \mapsto [0]$ , $[1] \mapsto [1]$ is a counterexample.

Facts & Assumptions

Given: The additive cyclic groups $C_{4} = Z / 4$ and $C_{2} = Z / 2$ .

[L1]

The false claim being refuted is the one stated in FALSE: a set-theoretic section of an extension is automatically a homomorphism.

[L2]

Cyclic groups are generated by one element, so the displayed additive models are valid presentations of $C_{4}$ and $C_{2}$ (Every cyclic group is isomorphic to $(Z, +)$ or to $(Z / n, +)$ for its finite order $n \geq 1$ ).

Counterexample

technique · direct

1.1

Reduction mod $2$ is a surjective homomorphism $π : C_{4} \to C_{2}$ , and the displayed map $s$ is a set-theoretic section because $π (s ([0])) = [0]$ and $π (s ([1])) = [1]$ .

givenL2L1

2.1

But $s ([1] + [1]) = s ([0]) = [0] \neq [2] = s ([1]) + s ([1])$ in $C_{4}$ . Hence $s$ is not a homomorphism. Therefore the universal claim [L1] is false.

L1step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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

J. S. Milne, Group Theory (standard reference, not scraped)