Norm (abelian group)

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

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

An alternative, stronger definition of a norm on [math]\displaystyle{ (G, +) }[/math] requires

1. [math]\displaystyle{ \nu(g) \gt 0 \text{ for all } g \ne e }[/math],
2. [math]\displaystyle{ \nu(g+h) \le \nu(g) + \nu(h) }[/math],
3. [math]\displaystyle{ \nu(mg) = |m| \, \nu(g) \text{ for all } m \in \mathbb{Z} }[/math].

The norm [math]\displaystyle{ \nu }[/math] is discrete if there is some real number [math]\displaystyle{ \rho \gt 0 }[/math] such that [math]\displaystyle{ \nu(g) \gt \rho }[/math] whenever [math]\displaystyle{ g \ne 0 }[/math].

Free abelian groups

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