perfect bilinear form

Let $A$, $B$, and $C$ be abelian groups. A bilinear form

$$\phi :A\times B\to C$$

is said to be if the associated group homomorphisms

$A$$\to \mathrm{Hom}(B,C)$

$a$$\mapsto \phi (a,\cdot )$

and

$B$$\to \mathrm{Hom}(A,C)$

$b$$\mapsto \phi (\cdot ,b)$

are injective.

In particular, if $C$ is finite then the finiteness of either $A$ or $B$ implies the finiteness of the other.