Contents

### Context

#### Algebra

higher algebra

universal algebra

group theory

# Contents

## Definition

A quadratic abelian group is an abelian group $G$ with a function $q: G \to \mathbb{Z}$ such that the following properties hold:

• (cube relation) For any $x,y,z \in G$,
$q(x+y+z) - q(x+y) - q(x+z) - q(y+z) + q(x) + q(y) + q(z) = 0$
• (homogeneous of degree 2) For any $x \in G$ and any $r \in \mathbb{Z}$,
$q(r x) = r^2 q(x)$

## Examples

• Every ring is a quadratic abelian group.

• Every inner product abelian group is a quadratic abelian group with $q(x) \coloneqq \langle x, x \rangle$.