Return to Question

A metric group is a finite abelian group $G$ with a quadratic function $$q:G\rightarrow \mathbb R/\mathbb Z\;,$$ that is, $$M(a,b):= q(a+b)-q(a)-q(b)$$ is bilinea in $a$ and $b$ [edit: and non-degenerate]. A Lagrangian subgroup $H\subset G$ is one with $q(h)=0$ $\forall h\in H$, and $|H|^2=|G|$.

Question: Is it true that $H$ is always isomorphic to $G/H$, and how can one see this? Can one somehow relate $G/H$ to the $\mathbb R/\mathbb Z$-valued representations of $H$ via $M$ (or vice versa)?

A metric group is a finite abelian group $G$ with a quadratic function $$q:G\rightarrow \mathbb R/\mathbb Z\;,$$ that is, $$M(a,b):= q(a+b)-q(a)-q(b)$$ is bilinear in $a$ and $b$ [edit: and non-degenerate]. A Lagrangian subgroup $H\subset G$ is one with $q(h)=0$ $\forall h\in H$, and $\lvert H\rvert^2=\lvert G\rvert$.

Question: Is it true that $H$ is always isomorphic to $G/H$, and how can one see this? Can one somehow relate $G/H$ to the $\mathbb R/\mathbb Z$-valued representations of $H$ via $M$ (or vice versa)?

A metric group is a finite abelian group $G$ with a quadratic function $$q:G\rightarrow \mathbb R/\mathbb Z\;,$$ that is, $$M(a,b):= q(a+b)-q(a)-q(b)$$ is bilinear in $a$ and $b$. A Lagrangian subgroup $H\subset G$ is one with $q(h)=0$ $\forall h\in H$, and $|H|^2=|G|$.

Question: Is it true that $H$ is always isomorphic to $G/H$, and how can one see this? Can one somehow relate $G/H$ to the $\mathbb R/\mathbb Z$-valued representations of $H$ via $M$ (or vice versa)?

A metric group is a finite abelian group $G$ with a quadratic function $$q:G\rightarrow \mathbb R/\mathbb Z\;,$$ that is, $$M(a,b):= q(a+b)-q(a)-q(b)$$ is bilinea in $a$ and $b$ [edit: and non-degenerate]. A Lagrangian subgroup $H\subset G$ is one with $q(h)=0$ $\forall h\in H$, and $|H|^2=|G|$.

Question: Is it true that $H$ is always isomorphic to $G/H$, and how can one see this? Can one somehow relate $G/H$ to the $\mathbb R/\mathbb Z$-valued representations of $H$ via $M$ (or vice versa)?

A metric group is a finite abelian group $G$ with a quadratic function $$q:G\rightarrow \mathbb R/\mathbb Z\;,$$ that is, $$M(a,b):= q(a+b)-q(a)-q(b)$$ is bilinear in $a$ and $b$. A Lagrangian subgroup $H\subset G$ is one with $q(h)=0 \forall h\in H$, and $|H|^2=|G|$.

Question: Is it true that $H$ is always isomorphic to $G/H$, and how can one see this? Can one somehow relate $G/H$ to the $\mathbb R/\mathbb Z$-valued representations of $H$ via $M$ (or vice versa)?

A metric group is a finite abelian group $G$ with a quadratic function $$q:G\rightarrow \mathbb R/\mathbb Z\;,$$ that is, $$M(a,b):= q(a+b)-q(a)-q(b)$$ is bilinear in $a$ and $b$. A Lagrangian subgroup $H\subset G$ is one with $q(h)=0$ $\forall h\in H$, and $|H|^2=|G|$.

Question: Is it true that $H$ is always isomorphic to $G/H$, and how can one see this? Can one somehow relate $G/H$ to the $\mathbb R/\mathbb Z$-valued representations of $H$ via $M$ (or vice versa)?