Abelian group
One of the most basic invariants of an infinite abelian group formula_1 is its rank: the cardinality of the maximal linearly independent subset of formula_1. Abelian groups of rank 0 are precisely the periodic groups, while torsion-free abelian groups of rank 1 are necessarily subgroups of formula_168 and can be completely described. More generally, a torsion-free abelian group of finite rank formula_158 is a subgroup of formula_203. On the other hand, the group of formula_73-adic integers formula_205 is a torsion-free abelian group of infinite formula_54-rank and the groups formula_207 with different formula_44 are non-isomorphic, so this invariant does not even fully capture properties of some familiar groups.