Product with Inverse on Homomorphic Image is Group Homomorphism/Mistake

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Source Work

1996: John F. Humphreys: A Course in Group Theory:

Chapter $8$: The Homomorphism Theorem:
Exercise $3$


Suppose that $H$ is an abelian group and let $\vartheta: G \to H$ be a homomorphism. Define a map $\phi: G \times G \to H$ by
$\map \phi {g_1, g_2} = \map \phi {g_1} \map \phi {g_2}^{-1}$.
Prove that $\phi$ is a homomorphism.


The condition is wrong.

It should read:

$\map \phi {g_1, g_2} = \map \vartheta {g_1} \map \vartheta {g_2}^{-1}$