CounterexampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-08-02

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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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The doubling endomorphism of $(Z, +)$ has trivial kernel but is not surjective

Statement refuted

An additive group homomorphism with trivial kernel must be surjective.

Facts & Assumptions

Given: The map $d : (Z, +) \to (Z, +)$ defined by $d (m) = 2 m$ .

[L1]

A group homomorphism is injective exactly when its kernel is trivial (A group homomorphism is injective if and only if its kernel is trivial).

[L2]

A group homomorphism preserves the group operation (Monoid homomorphism and group homomorphism).

[L3]

Surjectivity means that every codomain element is an image value (Injection, surjection, bijection).

[L4]

$Z$ is an additive group with cancellation (The integers form a commutative ring).

Counterexample

technique · direct

1.1

The equality $d (a + b) = 2 a + 2 b = d (a) + d (b)$ makes $d$ a group homomorphism.

L1L2L3L4givenalgebra

2.1

If $d (m) = 0$ , then $2 m = 0$ and integer cancellation gives $m = 0$ , so $\ker d = {0}$ .

step 1.1L1L2L3L4givenalgebra

3.1

But $1$ is not even and hence is not in the image of $d$ , refuting the stated implication.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

24 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

Ernst, An Inquiry-Based Approach to Abstract Algebra, Homomorphisms (standard reference, not scraped)

The doubling endomorphism of (Z,+)(Z,+) has trivial kernel but is not surjective