perfect bilinear form

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

$\varphi:A\times B\rightarrow C$

is said to be if the associated group homomorphisms

	$\displaystyle A$	$\displaystyle\rightarrow\operatorname{Hom}(B,C)$
	$\displaystyle a$	$\displaystyle\mapsto\varphi(a,\cdot)$

and

	$\displaystyle B$	$\displaystyle\rightarrow\operatorname{Hom}(A,C)$
	$\displaystyle b$	$\displaystyle\mapsto\varphi(\cdot,b)$

are injective.

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