In mathematics, two Prüfer theorems, named after Heinz Prüfer, describe the structure of certain infinite

abelian group In mathematics, 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 the order in which they are written. That is, the group operation is comm ...

s. 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 In abstract algebra, an abelian group (G,+) is called finitely generated if there exist finitely many elements x_1,\dots,x_s in G such that every x in G can be written in the form x = n_1x_1 + n_2x_2 + \cdots + n_sx_s for some integers n_1,\dots, n ...

, ''A'' is decomposable into a direct sum of

cyclic subgroup In abstract algebra, a generating set of a group is a subset of the group set such that every element of the group can be expressed as a combination (under the group operation) of finitely many elements of the subset and their inverses. In oth ...

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

isomorphism In mathematics, an isomorphism is a structure-preserving mapping 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 them. The word i ...

. 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 group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...

s. The second Prüfer theorem states that a

countable In mathematics, a set is countable if either it is 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 from it into the natural numbers ...

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 Union commonly refers to: * Trade union, an organization of workers * Union (set theory), in mathematics, a fundamental operation on sets Union may also refer to: Arts and entertainment Music * Union (band), an American rock group ** ''Un ...
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