mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, two Prüfer theorems, named after Heinz Prüfer, describe the structure of certain infinite abelian groups. They have been generalized by L. Ya. Kulikov.

Statement

Let ''A'' be an abelian group. If ''A'' is finitely generated then by the fundamental theorem of finitely generated abelian groups, ''A'' is decomposable into a

direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently but analogously for different kinds of structures. As an example, the direct sum of two abelian groups A and B is anothe ...

of cyclic

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

up to Two Mathematical object, mathematical objects and are called "equal up to an equivalence relation " * if and are related by , that is, * if holds, that is, * if the equivalence classes of and with respect to are equal. This figure of speech ...

isomorphism In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...

. 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 groups in two special cases. The first Prüfer theorem states that an abelian group of bounded exponent is isomorphic to a direct sum of cyclic groups. The second Prüfer theorem states that a

countable In mathematics, a Set (mathematics), set is countable if either it is finite set, finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function fro ...

abelian ''p''-group whose non-trivial elements have finite ''p''-height 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 ''A'' is isomorphic to a direct sum of cyclic groups if and only if it is a union of a sequence of subgroups with the property that the heights of all elements of ''A''_''i'' are bounded by a constant (possibly depending on ''i'').

References

* László Fuchs (1970), ''Infinite abelian groups, Vol. I''. Pure and Applied Mathematics, Vol. 36. New York–London: Academic Press * {{DEFAULTSORT:Prufer Theorems Abelian group theory Infinite group theory Theorems in group theory