# 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.

Title | perfect bilinear form |
---|---|

Canonical name | PerfectBilinearForm |

Date of creation | 2013-03-22 15:35:04 |

Last modified on | 2013-03-22 15:35:04 |

Owner | matsuura (2984) |

Last modified by | matsuura (2984) |

Numerical id | 5 |

Author | matsuura (2984) |

Entry type | Definition |

Classification | msc 15A63 |

Classification | msc 11E39 |