ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-08-11

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A pushout along an isomorphism recovers the other group

Example

Let $f : K \to G$ be an isomorphism and $h : K \to H$ any homomorphism. The pushout is $H$ , with legs $h \circ f^{- 1} : G \to H$ and ${i d}_{H}$ . For example, pushing $C_{4} \overset{≅}{\leftarrow} C_{4} \to C_{2}$ along reduction modulo $2$ gives $C_{2}$ .

Facts & Assumptions

Given: The objects and hypotheses in the example.

[L1]

Given homomorphisms $f : K \to G$ and $h : K \to H$ as in def-group-homomorphism, a pushout is a group $P$ with homomorphisms $i_{G} : G \to P$ and $i_{H} : H \to P$ such that $i_{G} \circ f = i_{H} \circ h$ , and such that every compatible pair $u : G \to Q$ , $v : H \to Q$ factors through a unique $w : P \to Q$ with $w \circ i_{G} = u$ and $w \circ i_{H} = v$ . The maps $f, h$ need not be injective. (Pushouts of group homomorphisms).

[L2]

Group isomorphisms, automorphisms and the set $Aut (G)$ . An isomorphism $f : G \to H$ is a bijective group homomorphism (def-group-homomorphism, def-injection-surjection-bijection). When $G = H$ , it is an automorphism of $G$ . Write $Aut (G) : = {f : G \to G : f is an automorphism} .$ (Group isomorphisms, automorphisms and the set $Aut (G)$ ).

[L3]

The inverse of a bijective group homomorphism is a group homomorphism. If $f : G \to H$ is a bijective group homomorphism, then its set-theoretic inverse $f^{- 1} : H \to G$ is a group homomorphism. (The inverse of a bijective group homomorphism is a group homomorphism).

[L4]

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 $(Z, +) / n Z$ ).

Verification

technique · direct

1.1

The two displayed legs agree on $K$ : $(h \circ f^{- 1}) \circ f = h$ .

givenL1L2L3L4

2.1

For a compatible pair $u : G \to Q$ , $v : H \to Q$ , one has $u = v \circ h \circ f^{- 1}$ , so $v$ is the unique mediator from $H$ .

step 1.1

3.1

In the cyclic example this says the $C_{4}$ leg is reduction modulo $2$ , the $C_{2}$ leg is the identity, and every compatible cocone factors uniquely through $C_{2}$ .

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

16 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.