# Product with Inverse on Homomorphic Image is Group Homomorphism/Mistake

## Source Work

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

## Mistake

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.

## Correction

The condition is wrong.

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