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

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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.

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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 $N ⊴ G$ , $f : G \to H$ is a homomorphism, and $N \subseteq \ker f$ , then there is a unique homomorphism $\overset{ˉ}{f} : G / N \to H$ such that $\overset{ˉ}{f} (g N) = f (g)$ and $f = \overset{ˉ}{f} \circ π$ .

Facts & Assumptions

Given: $N ⊴ G$ , a homomorphism $f : G \to H$ , and $N \subseteq \ker f$ .

[L1]

$G / N$ is the group of cosets of a normal subgroup (The quotient group $G / N$ and coset product $(g N) (h N) = g h N$ ).

[L2]

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

[L3]

$N \subseteq \ker f$ means that $f (n) = e_{H}$ for every $n \in N$ (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

1.1L1L2L3L4L5givenconstruct

Define $\overset{ˉ}{f} (g N) : = f (g)$ ; if $g N = h N$ , then $h^{- 1} g \in N \subseteq \ker f$ , so [L4] proves that this value is independent of the representative.

2.1step 1.1L1L2L3L4L5givenalgebra

For cosets, $\overset{ˉ}{f} ((g N) (h N)) = f (g h) = f (g) f (h)$ , and $f (g) = \overset{ˉ}{f} (π (g))$ .

3.1step 2.1∎

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

Depends on

Dependency tree · 2 levels

Sources

Milne, Group Theory, Kernels and Quotients