Prüfer group

The groups are named after Heinz Prüfer, a German mathematician of the early 20th century.

The Prüfer p-group may be identified with the subgroup of the circle group, U(1), consisting of all pn-th roots of unity as n ranges over all non-negative integers: The group operation here is the multiplication of complex numbers.

There is a presentation Here, the group operation in Z(p∞) is written as multiplication.

Alternatively and equivalently, the Prüfer p-group may be defined as the Sylow p-subgroup of the quotient group Q/Z, consisting of those elements whose order is a power of p: (where Z[1/p] denotes the group of all rational numbers whose denominator is a power of p, using addition of rational numbers as group operation).

For each natural number n, consider the quotient group Z/pnZ and the embedding Z/pnZ → Z/pn+1Z induced by multiplication by p. The direct limit of this system is Z(p∞): If we perform the direct limit in the category of topological groups, then we need to impose a topology on each of the

is a cyclic subgroup of Z(p∞) with pn elements; it contains precisely those elements of Z(p∞) whose order divides pn and corresponds to the set of pn-th roots of unity.

The Prüfer p-groups are the only infinite groups whose subgroups are totally ordered by inclusion.

This sequence of inclusions expresses the Prüfer p-group as the direct limit of its finite subgroups.

The Prüfer p-group is the unique infinite p-group that is locally cyclic (every finite set of elements generates a cyclic group).

The Prüfer p-groups are the only infinite abelian groups with this property.

More precisely: an abelian group is divisible if and only if it is the direct sum of a (possibly infinite) number of copies of Q and (possibly infinite) numbers of copies of Z(p∞) for every prime p. The (cardinal) numbers of copies of Q and Z(p∞) that are used in this direct sum determine the divisible group up to isomorphism.

[4] In the theory of locally compact topological groups the Prüfer p-group (endowed with the discrete topology) is the Pontryagin dual of the compact group of p-adic integers, and the group of p-adic integers is the Pontryagin dual of the Prüfer p-group.