In mathematics, two Prüfer theorems, named after Heinz Prüfer

Heinz Prüfer

Ernst Paul Heinz Prüfer was a German mathematician, who worked on abelian groups, algebraic numbers, knot theory and Sturm-Liouville theory. His advisor was Issai Schur.- See also :* Prüfer manifold...

, describe the structure of certain infinite abelian group

Abelian group

In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...

s. They have been generalized by L. Ya. Kulikov.

Statement

Let A be an abelian group. If A is finitely generated

Finitely generated abelian group

In abstract algebra, an abelian group is called finitely generated if there exist finitely many elements x1,...,xs in G such that every x in G can be written in the formwith integers n1,...,ns...

 then by the fundamental theorem of finitely generated abelian groups, A is decomposable into a direct sum

Direct sum

In mathematics, one can often define a direct sum of objectsalready known, giving a new one. This is generally the Cartesian product of the underlying sets , together with a suitably defined structure. More abstractly, the direct sum is often, but not always, the coproduct in the category in question...

 of cyclic subgroup

Cyclic group

In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...

s, which leads to the classification of finitely generated abelian groups up to isomorphism

Isomorphism

In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations.  If there exists an isomorphism between two structures, the two structures are said to be isomorphic.  In a certain sense, isomorphic structures are...

. The structure of general infinite abelian groups can be considerably more complicated and the conclusion needs not to hold, but Prüfer proved that it remains true for periodic group

Periodic group

In group theory, a periodic group or a torsion group is a group in which each element has finite order. All finite groups are periodic. The concept of a periodic group should not be confused with that of a cyclic group, although all finite cyclic groups are periodic.The exponent of a periodic group...

s in two special cases.

The first Prüfer theorem states that an abelian group of bounded exponent is isomorphic to a direct sum of cyclic group

Cyclic group

In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...

s. The second Prüfer theorem states that a countable

Countable set

In mathematics, a countable set is a set with the same cardinality as some subset of the set of natural numbers. A set that is not countable is called uncountable. The term was originated by Georg Cantor...

  periodic abelian group whose elements have finite height

Height (abelian group)

In mathematics, the height of an element g of an abelian group A is an invariant that captures its divisibility properties: it is the largest natural number N such that the equation Nx = g has a solution x ∈ A, or symbol ∞ if the largest number with this property does not exist...

 is isomorphic to a direct sum of cyclic groups. Examples show that the assumption that the group be countable cannot be removed.

The two Prüfer theorems follow from a general criterion of decomposability of an abelian group into a direct sum of cyclic subgroups due to L. Ya. Kulikov.

An abelian p-group

P-group

In mathematics, given a prime number p, a p-group is a periodic group in which each element has a power of p as its order: each element is of prime power order. That is, for each element g of the group, there exists a nonnegative integer n such that g to the power pn is equal to the identity element...

 A is isomorphic to a direct sum of cyclic groups if and only if it is a union

Union (set theory)

In set theory, the union of a collection of sets is the set of all distinct elements in the collection. The union of a collection of sets S_1, S_2, S_3, \dots , S_n\,\! gives a set S_1 \cup S_2 \cup S_3 \cup \dots \cup S_n.- Definition :...

of a sequence {Ai} of subgroups with the property that the heights of all elements of Ai are bounded by a constant (possibly depending on i).