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''group of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076440/q0764402.png" />''
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''Prüfer $p$-group, group of type $p^\infty$''
  
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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An infinite Abelian [[P-group|$p$-group]] all proper subgroups of which are [[Cyclic group|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
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$$
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z^{p^n} = 1,\ \ n=1,2,\ldots
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$$
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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 number|$p$-adic number]]s 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]]
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$$
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\lim_{\longrightarrow n} C_n
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$$
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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
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$$
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a_1^p = 1,\ \ a_{n+1}^p = a_{n},\ \ n=1,2,\ldots \ .
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$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076440/q0764405.png" /></td> </tr></table>
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Quasi-cyclic groups are the only infinite Abelian (and also the only [[Locally finite group|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.
  
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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Quasi-cyclic groups are [[Divisible group|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(2)|Frattini subgroup]].
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076440/q07644014.png" /></td> </tr></table>
 
 
 
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
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076440/q07644017.png" /></td> </tr></table>
 
 
 
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====
 
====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>
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<table>
 
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<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>
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<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>
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</table>
  
  
 
====Comments====
 
====Comments====
 
A quasi-cyclic group is better known as a Prüfer group in the West.
 
A quasi-cyclic group is better known as a Prüfer group in the West.
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Latest revision as of 20:17, 16 October 2017

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.

How to Cite This Entry:
Quasi-cyclic group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quasi-cyclic_group&oldid=14132
This article was adapted from an original article by N.N. Vil'yams (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article