Binate group

Definition with symbols
A group $$G$$ is termed a binate group if for every finitely generated subgroup $$H$$ of $$G$$ there is a homomorphism $$\varphi_H: H \to G$$ and an element $$u_H \in G$$ such that for all $$h \in H$$, we have:

$$h = [u_H, \varphi_H(h)]$$.

Weaker properties

 * Stronger than::Acyclic group: