TheoremStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-26

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Homomorphisms out of a central product

Statement

Let $G$ and $H$ be groups with central subgroups $Z_{1} \leq Z (G)$ and $Z_{2} \leq Z (H)$ and an isomorphism $α : Z_{1} \to Z_{2}$ , and let $K$ be a group. If $φ : G \to K$ and $ψ : H \to K$ are homomorphisms with commuting images and $φ ∣_{Z_{1}} = ψ \circ α$ , then there is a unique homomorphism $G \circ_{α} H \to K$ restricting to $φ$ and $ψ$ along the canonical maps. Explicitly it sends $\overset{ˉ}{g} \overset{ˉ}{h}$ to $φ (g) ψ (h)$ .

Facts & Assumptions

Given: Groups $G, H, K$ , central subgroups $Z_{1} \leq Z (G)$ and $Z_{2} \leq Z (H)$ , an isomorphism $α : Z_{1} \to Z_{2}$ , and homomorphisms $φ : G \to K$ , $ψ : H \to K$ with $φ (g) ψ (h) = ψ (h) φ (g)$ for all $g, h$ and $φ (z) = ψ (α (z))$ for all $z \in Z_{1}$ .

[F1]

The external direct product $G \times H : = {(g, h) : g \in G, h \in H}$ carries the componentwise operation $(g, h) (g^{'}, h^{'}) = (g g^{'}, h h^{'})$ (The external direct product $G \times H$ with componentwise multiplication).

[F2]

For groups $G, H$ with central subgroups $Z_{1} \leq Z (G)$ , $Z_{2} \leq Z (H)$ and an isomorphism $α : Z_{1} \to Z_{2}$ , the central product $G \circ_{α} H$ is the quotient of $G \times H$ by $N = {(z, α (z)^{- 1}) : z \in Z_{1}}$ (The central product $G \circ_{α} H$ of two groups along an isomorphism of central subgroups).

[F3]

For a group homomorphism $f : G \to H$ , $\ker f : = {g \in G : f (g) = e_{H}}$ (The kernel and image of a group homomorphism).

[F4]

The quotient group $G / N$ has the left cosets $g N$ as elements, with product $(g N) (h N) : = g h N$ (The quotient group $G / N$ and coset product $(g N) (h N) = g h N$ ).

[L1]

A group homomorphism $f$ satisfies $f (e) = e^{'}$ and $f (g^{- 1}) = f (g)^{- 1}$ (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).

[L2]

For $H \leq G$ the rule $(a H) (b H) : = a b H$ on left cosets is independent of the representatives $a$ and $b$ if and only if $H ⊴ G$ (Coset multiplication $(g H) (h H) = g h H$ is well defined if and only if $H$ is normal).

[L3]

The subgroup $N = {(z, α (z)^{- 1}) : z \in Z_{1}}$ of $G \times H$ is central, hence normal (The identified subgroup used to form a central product is central, hence normal).

[L4]

The canonical maps $G \to G \circ_{α} H$ and $H \to G \circ_{α} H$ are injective homomorphisms whose images commute elementwise, generate $G \circ_{α} H$ , and meet in the image of $Z_{1}$ (The two canonical maps into a central product are injective homomorphisms whose images commute, generate it, and meet in the identified centre).

Proof

technique · direct

1.1

The assignment $θ (g, h) : = φ (g) ψ (h)$ satisfies $θ ((g, h) (g^{'}, h^{'})) = φ (g) φ (g^{'}) ψ (h) ψ (h^{'}) = φ (g) ψ (h) φ (g^{'}) ψ (h^{'}) = θ (g, h) θ (g^{'}, h^{'})$ , the middle equality being the commuting-images hypothesis; so $θ : G \times H \to K$ is a homomorphism.

F1givenalgebra

1.2

For $z \in Z_{1}$ , $θ (z, α (z)^{- 1}) = φ (z) ψ (α (z)^{- 1}) = φ (z) ψ (α (z))^{- 1} = φ (z) φ (z)^{- 1} = e$ , so $N \leq \ker θ$ .

F2F3L1given

2.1

If $x N = y N$ in $G \times H$ then $x^{- 1} y \in N$ , so $θ (x)^{- 1} θ (y) = θ (x^{- 1} y) = e$ and $θ (x) = θ (y)$ ; hence $\overset{ˉ}{θ} (x N) : = θ (x)$ is a well-defined function on $G \circ_{α} H$ , and it is a homomorphism because $N$ is normal and $(x N) (y N) = x y N$ .

F4L2L3step 1.1step 1.2

3.1

On the canonical images, $\overset{ˉ}{θ} (\overset{ˉ}{g}) = θ (g, e) = φ (g)$ and $\overset{ˉ}{θ} (\overset{ˉ}{h}) = θ (e, h) = ψ (h)$ ; and any homomorphism agreeing with $φ$ and $ψ$ on the two images agrees with $\overset{ˉ}{θ}$ on a generating set of $G \circ_{α} H$ , hence everywhere.

L1L4step 2.1∎

Remarks

Both hypotheses are needed and neither is implied by the other. Without commuting images the assignment of step 1.1 is not a homomorphism on the direct product; without the agreement on $Z_{1}$ the subgroup $N$ need not lie in the kernel, so nothing descends to the quotient.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

M. van Beek, Topics in Finite p-Groups, §2.4 (standard reference, not scraped)
D. A. Craven, The Theory of p-Groups, §3.1 (standard reference, not scraped)