An abelian group where one could halve each element or divide each element by two.

Definition

A halving group is an abelian groupG with a function $(-)/2:G \to G$ called halving or dividing by two such that for all $g \in G$, $g/2 + g/2 = g$.

Properties

Just as every abelian group is a $\mathbb{Z}$-module, every halving group is a $\mathbb{D}$-module, where $\mathbb{D}$ are the dyadic rational numbers.

The only element of a halving group with order$2$ is the additive unit $0$.