# Quasihomomorphism of groups

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The term quasihomomorphism is used in a number of different contexts, many of them different from this one

WARNING: POTENTIAL TERMINOLOGICAL CONFUSION: Please don't confuse this with quasimorphism

## Contents

This article defines a term that has been used or referenced in a journal article or standard publication, but may not be generally accepted by the mathematical community as a standard term.[SHOW MORE]

## Definition

Let $G$ and $H$ be groups. A map $f:G \to H$ is termed a quasihomomorphism of groups if it satisfies the following equivalent conditions:

• Given any homomorphism $\varphi:A \to G$ from an abelian group $A$ to $G$, the composite $f \circ \varphi$ is a homomorphism from $A$ to $H$.
• If $a,b \in G$ commute, then $f(ab) = f(a)f(b)$.