LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-02

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.

The inverse of a bijective group homomorphism is a group homomorphism

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.

Given: A bijective group homomorphism $f : G \to H$ .

[L1]

[L2]

[L3]

A bijection has a two-sided set-theoretic inverse (Injection, surjection, bijection).

technique · direct

1.1L1L2L3givenchoose

By [L3], for $u, v \in H$ choose $x, y \in G$ with $u = f (x)$ and $v = f (y)$ ; then $u v = f (x y)$ by [L2].

2.1step 1.1L1L2L3givenalgebra

Therefore $f^{- 1} (u v) = x y = f^{- 1} (u) f^{- 1} (v)$ , so the inverse preserves the group operation.

3.1step 2.1∎

Hence $f^{- 1} : H \to G$ is a group homomorphism.

Depends on