CorollaryStatement: AI-generatedProof: 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 is isomorphic to the other factor

Statement

If $f : K \to G$ is an isomorphism and $h : K \to H$ is any homomorphism, then the pushout is isomorphic to $H$ , compatibly with the canonical maps.

Facts & Assumptions

Given: The objects and hypotheses in the statement.

[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).

Proof

technique · direct

1.1

The maps $h \circ f^{- 1} : G \to H$ and ${i d}_{H} : H \to H$ are compatible because $(h \circ f^{- 1}) \circ f = h$ .

givenL1L2L3

2.1

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

step 1.1

3.1

Thus $H$ satisfies the pushout universal property and is uniquely isomorphic to the given pushout. The proof also covers trivial groups.

step 2.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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