False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-11

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FALSE: canonical factor maps into every group pushout are injective

Statement

False claim: both canonical factor maps into every group pushout are injective.

Facts & Assumptions

Given: The objects and hypotheses in the statement.

[L1]

For a pushout of $f : K \to G$ and $h : K \to H$ , $i_{H} (h (\ker f)) = {e}, i_{G} (f (\ker h)) = {e} .$ Hence canonical factor maps in an arbitrary group pushout need not be injective. No equality with their full kernels is asserted. (The kernels of the amalgamating maps are killed in the opposite canonical maps to a group pushout).

[L2]

For homomorphisms $f : K \to G$ and $h : K \to H$ , let $N$ be the normal closure in $G * H$ of ${j_{G} (f (k)) j_{H} (h (k))^{- 1} : k \in K} .$ Then $(G * H) / N$ , with the induced factor maps $j_{G}$ and $j_{H}$ , is a pushout of $f$ and $h$ . (A group pushout is the quotient of a free product by the amalgamating relations).

[L3]

For every $n \in N$ , view $n$ as its canonical nonnegative integer and put $n Z : = {n k : k \in Z}$ . Then the left cosets of $n Z$ in $(Z, +)$ are exactly the congruence classes modulo $n$ , and coset addition is the published addition of congruence classes. Thus $(Z, +) / n Z = (Z / n, +)$ as the same group on the same underlying set. This includes $n = 0$ and $n = 1$ . (For every $n \in N$ , the congruence-class group $(Z / n, +)$ is the quotient group $(Z, +) / n Z$ ).

Refutation

technique · direct

1.1

Take $K = C_{2}$ , $G = 1$ , $H = C_{2}$ , with $f : K \to G$ trivial and $h : K \to H$ the identity.

givenL1L2L3

2.1

The amalgamating relation kills the generator of $H$ , so the quotient construction makes the pushout trivial. Equivalently, kernel collapse kills $h (\ker f) = H$ .

step 1.1

3.1

The canonical map $H = C_{2} \to 1$ is not injective, refuting the claim.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

19 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

George D. Torres, Combinatorial Group Theory, §2 (standard reference, not scraped)
B. H. Neumann, Lectures on Topics in the Theory of Infinite Groups, Ch. 9 (standard reference, not scraped)