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TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02
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A homomorphism that kills a normal subgroup factors uniquely through the quotient group

Statement

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

If N⊴GN\mathrel{\trianglelefteq}G, f:G→Hf:G\to H is a homomorphism, and N⊆ker⁡fN\subseteq\ker f, then there is a unique homomorphism fˉ:G/N→H\bar f:G/N\to H such that fˉ(gN)=f(g)\bar f(gN)=f(g) and f=fˉ∘πf=\bar f\circ\pi.

Facts & Assumptions

Given: N⊴GN\mathrel{\trianglelefteq}G, a homomorphism f:G→Hf:G\to H, and N⊆ker⁡fN\subseteq\ker f.

[L1]

G/NG/N is the group of cosets of a normal subgroup (The quotient group G/NG/N and coset product (gN)(hN)=ghN(gN)(hN)=ghN).

[L2]

The quotient map π(g)=gN\pi(g)=gN is a surjective homomorphism (The canonical projection π:G→G/N\pi:G\to G/N, π(g)=gN\pi(g)=gN, is a surjective group homomorphism).

[L3]

N⊆ker⁡fN\subseteq\ker f means that f(n)=eHf(n)=e_H for every n∈Nn\in N (The kernel and image of a group homomorphism).

[L4]

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

[L5]

A group homomorphism preserves products (Monoid homomorphism and group homomorphism).

Proof

technique · constructive
1.1

Define fˉ(gN):=f(g)\bar f(gN):=f(g); if gN=hNgN=hN, then h−1g∈N⊆ker⁡fh^{-1}g\in N\subseteq\ker f, so [L4] proves that this value is independent of the representative.

L1L2L3L4L5givenconstruct
2.1

For cosets, fˉ((gN)(hN))=f(gh)=f(g)f(h)\bar f((gN)(hN))=f(gh)=f(g)f(h), and f(g)=fˉ(π(g))f(g)=\bar f(\pi(g)).

step 1.1L1L2L3L4L5givenalgebra
3.1

The surjectivity used in step 2.1 forces any such factor map to have these values, hence proves uniqueness.

step 2.1∎

Depends on

Used by

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