# Quasi-cyclic group

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Prüfer $p$-group, group of type $p^\infty$
An infinite Abelian $p$-group all proper subgroups of which are cyclic. There exists for each prime number $p$ a quasi-cyclic group, and it is unique up to an isomorphism. This group is isomorphic to the multiplicative group of all roots of the equations $$z^{p^n} = 1,\ \ n=1,2,\ldots$$ in the field of complex numbers with the usual multiplication, and also to the quotient group $\mathbf{Q}_p/\mathbf{Z}_p$, where $\mathbf{Q}_p$ is the additive group of the field of rational $p$-adic numbers and $\mathbf{Z}_p$ is the additive group of the ring of all $p$-adic integers. A quasi-cyclic group is the union of an ascending chain of cyclic groups $C_n$ of orders $p^n$, $n=1,2,\ldots$; more precisely, it is the inductive limit $$\lim_{\longrightarrow n} C_n$$ with respect to the inductive system $(C_n,\phi_n)$. This group can be defined in terms of generators and relations as the group with countable system of generators $a_1,a_2,\ldots$ and relations $$a_1^p = 1,\ \ a_{n+1}^p = a_{n},\ \ n=1,2,\ldots \ .$$
Quasi-cyclic groups are divisible Abelian groups, and each divisible Abelian group is the direct sum of a set of groups that are isomorphic to the additive group of rational numbers and to quasi-cyclic groups for certain prime numbers $p$. Groups of type $p^\infty$ are maximal $p$-subgroups of the multiplicative group of complex numbers, and also maximal $p$-subgroups of the additive group of rational numbers modulo 1. The ring of endomorphisms of a group of type $p^\infty$ is isomorphic to the ring of $p$-adic integers. A quasi-cyclic group coincides with its Frattini subgroup.