Binate group

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Definition

Definition with symbols

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

$h=[u_H,{\phi }_H(h)]$.

Relation with other properties

Weaker properties

• Acyclic group: For proof of the implication, refer Binate implies acyclic and for proof of its strictness (i.e. the reverse implication being false) refer Acyclic not implies binate.