Norm (abelian group)

In mathematics, specifically abstract algebra, if $(G,+)$ is an (abelian) group with identity element $e$ then $\nu :G\to \mathbb{ℝ}$ is said to be a norm on $(G,+)$ if:

1. Positive definiteness: $\nu (g)>0\text{ for all }g\ne e\text{ and }\nu (e)=0$,
2. Subadditivity: $\nu (g+h)\le \nu (g)+\nu (h)$,
3. Inversion (Symmetry): $\nu (-g)=\nu (g)\text{ for all }g\in G$.

An alternative, stronger definition of a norm on $(G,+)$ requires

1. $\nu (g)>0\text{ for all }g\ne e$,
2. $\nu (g+h)\le \nu (g)+\nu (h)$,
3. $\nu (mg)=|m|\nu (g)\text{ for all }m\in \mathbb{\mathbb{Z}}$.

The norm $\nu$ is discrete if there is some real number $\rho >0$ such that $\nu (g)>\rho$ whenever $g\ne 0$.

Free abelian groups

An abelian group is a free abelian group if and only if it has a discrete norm.