Quasi-cyclic group

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 the only infinite Abelian (and also the only locally-finite infinite) groups all subgroups of which are finite. The question of the existence of infinite non-Abelian groups with this property is still unsolved (1978) and constitutes one of the problems of O.Yu. Shmidt.

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.

References

[1]	A.G. Kurosh, "The theory of groups" , 1–2 , Chelsea (1955–1956) (Translated from Russian)
[2]	M.I. Kargapolov, J.I. [Yu.I. Merzlyakov] Merzljakov, "Fundamentals of the theory of groups" , Springer (1979) (Translated from Russian)

Comments

A quasi-cyclic group is better known as a Prüfer group in the West.