1-homomorphism of groups

Definition
Let $$G$$ and $$H$$ be groups. A map $$f:G \to H$$ is termed a 1-homomorphism if it satisfies the following equivalent conditions:


 * For any element $$g \in G$$, the restriction of $$f$$ to the cyclic subgroup generated by $$g$$, is a homomorphism of groups.
 * For any homomorphism from a cyclic group to $$G$$, the composite with $$f$$ is also a homomorphism.

Stronger properties

 * Semihomomorphism of groups
 * Quasihomomorphism of groups
 * Homomorphism of groups