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.

Relation with other properties

Stronger properties

• Semihomomorphism of groups
• Quasihomomorphism of groups
• Homomorphism of groups