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Difference between revisions of "Torsion group"

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A [[Group|group]] in which every element has finite order. Any periodic [[Abelian group|Abelian group]] splits into a direct sum of primary groups with respect to distinct prime numbers. See [[Burnside problem|Burnside problem]] on periodic groups for finiteness conditions of periodic groups.
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A ''torsion group'' (also called ''periodic group'')
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is a
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[[Group|group]] in which every element has finite order. Any torsion
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[[Abelian group|Abelian group]] splits into a direct sum of primary groups with respect to distinct prime numbers. See
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[[Burnside problem|Burnside problem]] on torsion groups for finiteness conditions of torsion groups.
  
  
  
 
====Comments====
 
====Comments====
A periodic group is also called a torsion group. For any group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072180/p0721801.png" />, its torsion subgroup is defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072180/p0721802.png" />. It is a normal subgroup and the quotient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072180/p0721803.png" /> is the torsion-free quotient group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072180/p0721804.png" />. Both <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072180/p0721805.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072180/p0721806.png" /> are functors.
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For any group $G$, its torsion subgroup is defined by $T(G) = \{ g\in G \;|\; \exists n\in\N \textrm{ such that } g^n = e\}$. It is a normal subgroup and the quotient $F(G) = G/T(G)$ is the torsion-free quotient group of $G$. Both $T(\;.\;)$ and $F(\;.\;)$ are functors.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</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">[a2]</TD> <TD valign="top"> P. Hall,  "The theory of groups" , Macmillan (1959pp. Chapt. 10</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> D.J.S. Robinson,  "Finiteness condition and generalized soluble groups" , '''I''' , Springer  (1972)</TD></TR></table>
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|valign="top"|{{Ref|Ha}}||valign="top"| P. Hall,  "The theory of groups", Macmillan (1959pp. Chapt. 10  {{MR|0103215}}  {{ZBL|0084.02202}}
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|valign="top"|{{Ref|Ku}}||valign="top"| A.G. Kurosh,  "The theory of groups", '''1–2''', Chelsea  (1955–1956) (Translated from Russian{{MR|0071422}}  {{ZBL|0111.02502}}
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|valign="top"|{{Ref|Ro}}||valign="top"| D.J.S. Robinson,  "Finiteness condition and generalized soluble groups", '''I''', Springer  (1972) {{MR|0332990}} {{MR|0332989}} 
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Latest revision as of 21:20, 29 April 2012

2020 Mathematics Subject Classification: Primary: 20-XX [MSN][ZBL]

A torsion group (also called periodic group) is a group in which every element has finite order. Any torsion Abelian group splits into a direct sum of primary groups with respect to distinct prime numbers. See Burnside problem on torsion groups for finiteness conditions of torsion groups.


Comments

For any group $G$, its torsion subgroup is defined by $T(G) = \{ g\in G \;|\; \exists n\in\N \textrm{ such that } g^n = e\}$. It is a normal subgroup and the quotient $F(G) = G/T(G)$ is the torsion-free quotient group of $G$. Both $T(\;.\;)$ and $F(\;.\;)$ are functors.

References

[Ha] P. Hall, "The theory of groups", Macmillan (1959) pp. Chapt. 10 MR0103215 Zbl 0084.02202
[Ku] A.G. Kurosh, "The theory of groups", 1–2, Chelsea (1955–1956) (Translated from Russian) MR0071422 Zbl 0111.02502
[Ro] D.J.S. Robinson, "Finiteness condition and generalized soluble groups", I, Springer (1972) MR0332990 MR0332989
How to Cite This Entry:
Torsion group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Torsion_group&oldid=11928
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article