View source for Quasi-cyclic group

''group of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076440/q0764402.png" />''

An infinite Abelian [[P-group|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076440/q0764403.png" />-group]] all proper subgroups of which are cyclic (cf. [[Cyclic group|Cyclic group]]). There exists for each prime number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076440/q0764404.png" /> 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

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in the field of complex numbers with the usual multiplication, and also to the quotient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076440/q0764406.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076440/q0764407.png" /> is the additive group of the field of rational <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076440/q0764408.png" />-adic numbers and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076440/q0764409.png" /> is the additive group of the ring of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076440/q07644010.png" />-adic integers. A quasi-cyclic group is the union of an ascending chain of cyclic groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076440/q07644011.png" /> of orders <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076440/q07644012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076440/q07644013.png" />; more precisely, it is the inductive limit

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with respect to the inductive system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076440/q07644015.png" />. This group can be defined in terms of generators and relations as the group with countable system of generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076440/q07644016.png" /> and relations

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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 (cf. [[Divisible group|Divisible group]]), 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076440/q07644018.png" />. Groups of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076440/q07644019.png" /> are maximal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076440/q07644020.png" />-subgroups of the multiplicative group of complex numbers, and also maximal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076440/q07644021.png" />-subgroups of the additive group of rational numbers modulo 1. The ring of endomorphisms of a group of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076440/q07644022.png" /> is isomorphic to the ring of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076440/q07644023.png" />-adic integers. A quasi-cyclic group coincides with its [[Frattini-subgroup(2)|Frattini subgroup]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.G. Kurosh,   "The theory of groups" , '''1–2''' , Chelsea  (1955–1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M.I. Kargapolov,   J.I. [Yu.I. Merzlyakov] Merzljakov,   "Fundamentals of the theory of groups" , Springer  (1979)  (Translated from Russian)</TD></TR></table>



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