Norm (abelian Group)

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

#Positive definiteness: $\nu(g) > 0 \text g \ne e \text \nu(e) = 0$,
#Subadditivity: $\nu(g+h) \le \nu(g) + \nu(h)$,
#Inversion (Symmetry): $\nu(-g) = \nu(g) \text g \in G$.

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

#$\nu(g) > 0 \text g \ne e$,
#$\nu(g+h) \le \nu(g) + \nu(h)$,
#$\nu(mg) = , m, \, \nu(g) \text m \in \mathbb$.

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.

References

Abelian group theory

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