This is the Prüfer p-group.

The complete list of subgroups of the Prüfer p-group Z(p) is:

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

As there is no maximal subgroup of a Prüfer p-group, it is its own Frattini subgroup.

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

The Prüfer p-group may be identified with the subgroup of the circle group, U(1), consisting of all p-th roots of unity as n ranges over all non-negative integers:

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:

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.