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

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FALSE: a set-theoretic section of an extension is automatically a homomorphism

Statement

Every set-theoretic section of the surjection in a group extension is a group homomorphism.

Facts & Assumptions

Given: The quotient map $π : C_{4} \to C_{2}$ written additively, and the set section $s ([0]) = [0]$ , $s ([1]) = [1]$ .

[L1]

A split extension requires a homomorphic section, not merely a set section (Group extensions, sections, complements, and split extensions).

[L2]

A group homomorphism must preserve addition in cyclic additive notation (Monoid homomorphism and group homomorphism).

Refutation

technique · direct

1.1

The map $s$ is a set-theoretic section because reducing mod $2$ sends $[0]$ to $[0]$ and $[1]$ to $[1]$ .

givenalgebra

2.1

In $C_{2}$ we have $[1] + [1] = [0]$ , but in $C_{4}$ we get $s ([1] + [1]) = s ([0]) = [0] \neq [2] = [1] + [1] = s ([1]) + s ([1])$ . So [L2] shows that $s$ is not a homomorphism. Therefore the claim is false, and [L1] explains why set sections alone do not split extensions.

L1L2step 1.1∎

Depends on

Used by

A set-theoretic section of C₄ onto C₂ need not be a homomorphism Counterexample

Dependency tree · two levels

8 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)