In mathematics, specifically in group theory, the Prüfer p-group or the p-quasicyclic group or p∞-group, Z(p∞), for a prime number p is the unique p-group in which every element has p different p-th roots. The Prüfer p-groups are countable abelian groups that are important in the classification of infinite abelian groups: they (along with the group of rational numbers) form the smallest building blocks of all divisible groups. The groups are named after Heinz Prüfer, a German mathematician of the early 20th century. Constructions of Z(p∞) 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 , and take the final topology on . If we wish for to be Hausdorff, we must impose the discrete topology on each of the , resulting in to have the discrete topology. We can also write where Qp denotes the additive group of p-adic numbers and Zp is the subgroup of p-adic integers. Properties The complete list of subgroups of the Prüfer p-group Z(p∞) = Z[1/p]/Z is: (Here 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. As there is no maximal subgroup of a Prüfer p-group, it is its own Frattini subgroup. Given this list of subgroups, it is clear that the Prüfer