Prüfer theorems

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 of cyclic subgroups, which leads to the classification of finitely generated abelian groups up to isomorphism. 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 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:

References

• László Fuchs (1970), Infinite abelian groups, Vol. I. Pure and Applied Mathematics, Vol. 36. New York–London: Academic Press