the groups (periodic) and (torsion-free). Torsion groups An abelian group is called periodic or torsion, if every element has finite order. A direct sum of finite cyclic groups is periodic. Although the converse statement is not true in general, some special cases are known. The first and second Prüfer theorems state that if is a periodic group, and it either has a bounded exponent, i.e., for some natural number , or is countable and the -heights of the elements of are finite for each , then is isomorphic to a direct sum of finite cyclic groups. The cardinality of the set of direct summands isomorphic to in such a decomposition is an invariant of . These theorems were later subsumed in the Kulikov criterion. In a different direction, Helmut Ulm found an extension of the second Prüfer theorem to countable abelian -groups with elements of infinite height: those groups are completely classified by means of their Ulm invariants. Torsion-free and mixed groups An abelian group is called torsion-free if every non-zero element has infinite order. Several classes of torsion-free abelian groups have been studied extensively: Free abelian groups, i.e. arbitrary direct sums of Cotorsion and algebraically compact torsion-free groups such as the -adic integers Slender groups An abelian group that is neither periodic nor torsion-free is called mixed. If is an abelian group and is its torsion subgroup, then the factor group is torsion-free. However, in general the torsion subgroup is not a direct summand of , so is not isomorphic to . Thus the theory of mixed groups involves more than simply combining the results about periodic and torsion-free groups. The additive group of integers is torsion-free -module. Invariants and classification One of the most basic invariants of an infinite abelian group is its rank: the cardinality of the maximal linearly independent subset of . Abelian groups of rank 0 are precisely the periodic groups, while torsion-free abelian groups of rank 1 are necessarily subgroups of and can be completely described. More generally, a torsion-free abelian group of finite rank is a subgroup of . On the other hand, the group of -adic integers is a torsion-free abelian group of infinite -rank and the groups with different are non-isomorphic, so this invariant does not even fully capture properties of some familiar groups. The classification theorems for finitely generated, divisible, countable periodic, and rank 1 torsion-free abelian groups explained above