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

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

An alternative, stronger definition of a norm on ${\displaystyle (G,+)}$ requires

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

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

Free abelian groups

An abelian group is a free abelian group if and only if it has a discrete norm.^[2]

References