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CounterexampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02
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The doubling endomorphism of (Z,+)(\mathbb 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,+)→(Z,+)d:(\mathbb Z,+)\to(\mathbb Z,+) defined by d(m)=2md(m)=2m.

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

Counterexample

technique · direct
1.1

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

L1L2L3L4givenalgebra
2.1

If d(m)=0d(m)=0, then 2m=02m=0 and integer cancellation gives m=0m=0, so ker⁡d={0}\ker d=\{0\}.

step 1.1L1L2L3L4givenalgebra
3.1

But 11 is not even and hence is not in the image of dd, refuting the stated implication.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 45 results over 15 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources