fact, the modules over can be identified with the abelian groups. Theorems about abelian groups (i.e. modules over the principal ideal domain ) can often be generalized to theorems about modules over an arbitrary principal ideal domain. A typical example is the classification of finitely generated abelian groups which is a specialization of the structure theorem for finitely generated modules over a principal ideal domain. In the case of finitely generated abelian groups, this theorem guarantees that an abelian group splits as a direct sum of a torsion group and a free abelian group. The former may be written as a direct sum of finitely many groups of the form for prime, and the latter is a direct sum of finitely many copies of . If are two group homomorphisms between abelian groups, then their sum , defined by , is again a homomorphism. (This is not true if is a non-abelian group.) The set of all group homomorphisms from to is therefore an abelian group in its own right. Somewhat akin to the dimension of vector spaces, every abelian group has a rank. It is defined as the maximal cardinality of a set of linearly independent (over the integers) elements of the group. Finite abelian groups and torsion groups have rank zero, and every abelian group of rank zero is a torsion group. The integers and the rational numbers have rank one, as well as every nonzero additive subgroup of the rationals. On the other hand, the multiplicative group of the nonzero rationals has an infinite rank, as it is a free abelian group with the set of the prime numbers as a basis (this results from the fundamental theorem of arithmetic). The center of a group is the set of elements that commute with every element of . A group is abelian if and only if it is equal to its center . The center of a group is always a characteristic abelian subgroup of . If the quotient group of a group by its center is cyclic then is abelian. Finite abelian groups Cyclic groups of integers modulo , , were among the first examples of groups. It turns out that an arbitrary finite abelian group is isomorphic to a direct sum of finite cyclic groups of prime power order, and these orders are uniquely determined, forming a complete system of invariants. The automorphism group of a