quasicyclic

Quasicyclic, also known as the Prüfer p-group, is an infinite abelian p-group denoted Z(p^∞) or C_{p^∞}. It is the union of the cyclic groups of order p, p^2, p^3, and so on, with the natural inclusions, making it a direct limit of these cyclic groups. Equivalently, it can be realized as the subgroup of the circle group consisting of all p^n-th roots of unity for all n, or additively as the subgroup of Q/Z consisting of rationals whose denominators are powers of p.

The group has several defining properties. It is divisible and torsion, and every finite subgroup is cyclic. For each n ≥ 1 there is a unique subgroup of order p^n, and the proper subgroups are exactly these finite cyclic subgroups. It is countable and locally cyclic, meaning every finitely generated subgroup is cyclic.

Endomorphism and automorphism structures are well understood. The endomorphism ring End(Z(p^∞)) is isomorphic to the ring of p-adic integers Z_p, and the automorphism group Aut(Z(p^∞)) is isomorphic to the group of units Z_p^×.

In the broader context of abelian group theory, Z(p^∞) serves as the canonical example of a divisible p-group and appears as the p-primary component of Q/Z. It is unique up to isomorphism among infinite divisible p-groups and plays a key role in the classification of divisible groups and in illustrating properties of quasicyclic groups.