Quasihomomorphism of groups

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

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 $$f(ab) = f(a)f(b)$$.

Stronger properties

 * Homomorphism of groups

Weaker properties

 * 1-homomorphism of groups