Norm (abelian group)

For the Baer norm (intersection of normalizers), see Norm (group).

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

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

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

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

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

Free abelian groups

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