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

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The inverse of a bijective group homomorphism is a group homomorphism

Statement

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.

Facts & Assumptions

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

[L1]

An isomorphism is a bijective group homomorphism (Group isomorphisms, automorphisms and the set $Aut (G)$ ).

[L2]

A group homomorphism preserves products, identities, and inverses (A group homomorphism automatically satisfies $f (e) = e^{'}$ and $f (g^{- 1}) = f (g)^{- 1}$ , and $f (g^{n}) = f (g)^{n}$ for every $n \in Z$ ; for monoid homomorphisms preservation of the identity must be assumed).

[L3]

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

Proof

technique · direct

1.1

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].

L1L2L3givenchoose

2.1

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

step 1.1L1L2L3givenalgebra

3.1

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

step 2.1∎

Depends on

Used by

A pushout along an isomorphism is isomorphic to the other factor Corollary
Lie-group homomorphism, isomorphism, and automorphism Definition
A pushout along an isomorphism recovers the other group Example
Inn(G) is a normal subgroup of Aut(G) Theorem
The automorphisms of a group form a group under composition Theorem

Dependency tree · two levels

20 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

Ernst, An Inquiry-Based Approach to Abstract Algebra, Homomorphisms (standard reference, not scraped)