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.

Title perfect bilinear form PerfectBilinearForm 2013-03-22 15:35:04 2013-03-22 15:35:04 matsuura (2984) matsuura (2984) 5 matsuura (2984) Definition msc 15A63 msc 11E39