Gruppennorm

Gruppennorm is a concept in group theory referring to a function that assigns a nonnegative size to each element of a group in a way that mirrors the familiar notion of a norm on vector spaces, but adapted to the group operation. Formally, for a group G with identity e, a Gruppennorm is a function ||·||: G → [0, ∞) satisfying:

- ||e|| = 0

- ||g|| > 0 for all g ≠ e (positive definiteness; in some contexts this is stated as ||g|| ≥ 0 with equality only at e)

- ||g^{-1}|| = ||g||

- ||gh|| ≤ ||g|| + ||h|| for all g, h ∈ G (subadditivity)

From a Gruppennorm one obtains a left-invariant metric d(g,h) = ||g^{-1}h||, which makes G a metric space and often a topological group when the norm is compatible with the group operations.

Common constructions include the word length norm with respect to a generating set S: ||g|| is the minimal length of a word in S ∪ S^{-1} representing g. Another example is the trivial (discrete) norm, which yields the discrete metric. In geometric group theory, proper norms (where balls of radius n contain finitely many elements) are of particular interest, as they relate to growth and geometry of the group.

Variants include pseudonorms, where positivity is not guaranteed for all non-identity elements, and non-Archimedean norms, which satisfy a stronger triangle inequality. Gruppennormen thus provide a flexible framework for measuring group elements, connecting algebraic structure with geometric and topological properties.