nLab quadratic abelian group

Contents

Context

Algebra

Group theory

Contents

Definition

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

  • (cube relation) For any x,y,zGx,y,z \in G,
    q(x+y+z)q(x+y)q(x+z)q(y+z)+q(x)+q(y)+q(z)=0q(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 xGx \in G and any rr \in \mathbb{Z},
    q(rx)=r 2q(x)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)x,xq(x) \coloneqq \langle x, x \rangle.

 See also

Last revised on May 11, 2022 at 08:30:34. See the history of this page for a list of all contributions to it.