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TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02
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First isomorphism theorem for groups: G/ker⁡f≅im⁡fG/\ker f\cong\operatorname{im}f

Statement

First isomorphism theorem for groups: G/ker⁡f≅im⁡fG/\ker f\cong\operatorname{im}f.

For every homomorphism f:G→Hf:G\to H, the rule gker⁡f↦f(g)g\ker f\mapsto f(g) is an isomorphism from G/ker⁡fG/\ker f onto im⁡f\operatorname{im}f.

Facts & Assumptions

Given: A group homomorphism f:G→Hf:G\to H.

[L1]

A homomorphism killing a normal subgroup factors uniquely through the quotient (A homomorphism that kills a normal subgroup factors uniquely through the quotient group).

[L2]

ker⁡f\ker f is normal and im⁡f\operatorname{im}f is a subgroup (The image of a group homomorphism is a subgroup and its kernel is a normal subgroup).

[L3]

A homomorphism is injective exactly when its kernel is trivial (A group homomorphism is injective if and only if its kernel is trivial).

[L4]

Equal images are exactly equal kernel cosets (Two elements have the same image under a homomorphism if and only if they lie in the same coset of its kernel).

[L5]

An isomorphism is a bijective group homomorphism (Group isomorphisms, automorphisms and the set Aut⁡(G)\operatorname{Aut}(G)).

Proof

technique · direct
1.1

By [L2] and [L1], fˉ:G/ker⁡f→im⁡f\bar f:G/\ker f\to\operatorname{im}f, fˉ(gker⁡f)=f(g)\bar f(g\ker f)=f(g), is a well-defined homomorphism; [L4] also gives representative independence directly.

L1L2L3L4L5givenconstruct
2.1

Its image is all of im⁡f\operatorname{im}f, and fˉ(gker⁡f)=eH\bar f(g\ker f)=e_H implies f(g)=eHf(g)=e_H, hence gker⁡f=ker⁡fg\ker f=\ker f; therefore its kernel is trivial.

step 1.1L1L2L3L4L5givenalgebra
3.1

The trivial-kernel conclusion of step 2.1 makes fˉ\bar f an isomorphism.

step 2.1∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 37 results over 14 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources