Prüfer group
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 "p"-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/"p"Z and the embedding Z/"p"Z → Z/"p"Z induced by multiplication by "p". The direct limit of this system is Z("p"): We can also write where Q denotes the additive group of "p"-adic numbers and Z 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 formula_7 is a cyclic subgroup of Z("p") with "p" elements; it contains precisely those elements of Z("p") whose order divides "p" and corresponds to the set of "p"-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 "p"-groups are indecomposable (cannot be written as a direct sum of proper subgroups). More is true: the Prüfer "p"-groups are subdirectly irreducible. An abelian group is subdirectly irreducible if and only if it is isomorphic to a finite cyclic "p"-group or to a Prüfer group. The Prüfer "p"-group is the unique  infinite "p"-group that is locally cyclic (every finite set of elements generates a cyclic group). As seen above, all proper subgroups of Z("p") are finite. The Prüfer "p"-groups are the only infinite abelian groups with this property. The Prüfer "p"-groups are divisible. They play an important role in the classification of divisible groups; along with the rational numbers they are the simplest divisible groups. 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. As an abelian group (that is, as a Z-module), Z("p") is Artinian but not Noetherian. It can thus be used as a counterexample against the idea that every Artinian module is Noetherian (whereas every Artinian "ring" is Noetherian). The endomorphism ring of Z("p") is isomorphic to the ring of "p"-adic integers Z. 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.