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Difference between revisions of "Group without torsion"

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''torsion-free group''
 
''torsion-free group''
  
A group without non-trivial elements of finite order. Free, free solvable, free nilpotent, and free Abelian groups are torsion-free groups (cf. [[Free Abelian group|Free Abelian group]]; [[Free group|Free group]]; [[Nilpotent group|Nilpotent group]]; [[Solvable group|Solvable group]]). Direct, complete direct and free products (cf. [[Direct product|Direct product]]; [[Free product of groups|Free product of groups]]) of torsion-free groups are torsion-free. The quotient group of a torsion-free group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045350/g0453501.png" /> by a normal subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045350/g0453502.png" /> is a torsion-free group if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045350/g0453503.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045350/g0453504.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045350/g0453505.png" /> and for any natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045350/g0453506.png" />. An extension of a torsion-free group by a torsion-free group is a torsion-free group. If a group is residually a finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045350/g0453507.png" />-group (cf. [[Residually-finite group|Residually-finite group]]; [[P-group|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045350/g0453508.png" />-group]]) with respect to two different prime numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045350/g0453509.png" />, then it is a torsion-free group.
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A group without non-trivial elements of finite order. Free, free solvable, free nilpotent, and free Abelian groups are torsion-free groups (cf. [[Free Abelian group|Free Abelian group]]; [[Free group|Free group]]; [[Nilpotent group|Nilpotent group]]; [[Solvable group|Solvable group]]). Direct, complete direct and free products (cf. [[Direct product|Direct product]]; [[Free product of groups|Free product of groups]]) of torsion-free groups are torsion-free. The quotient group of a torsion-free group $G$ by a normal subgroup $H$ is a torsion-free group if and only if $x^n\in H$ implies $x\in H$ for all $x\in G$ and for any natural number $n$. An extension of a torsion-free group by a torsion-free group is a torsion-free group. If a group is residually a finite $p$-group (cf. [[Residually-finite group|Residually-finite group]]; [[P-group|$p$-group]]) with respect to two different prime numbers $p$, then it is a torsion-free group.
  
 
====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></table>
 
<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></table>

Latest revision as of 11:51, 26 April 2014

torsion-free group

A group without non-trivial elements of finite order. Free, free solvable, free nilpotent, and free Abelian groups are torsion-free groups (cf. Free Abelian group; Free group; Nilpotent group; Solvable group). Direct, complete direct and free products (cf. Direct product; Free product of groups) of torsion-free groups are torsion-free. The quotient group of a torsion-free group $G$ by a normal subgroup $H$ is a torsion-free group if and only if $x^n\in H$ implies $x\in H$ for all $x\in G$ and for any natural number $n$. An extension of a torsion-free group by a torsion-free group is a torsion-free group. If a group is residually a finite $p$-group (cf. Residually-finite group; $p$-group) with respect to two different prime numbers $p$, then it is a torsion-free group.

References

[1] A.G. Kurosh, "The theory of groups" , 1–2 , Chelsea (1955–1956) (Translated from Russian)
How to Cite This Entry:
Group without torsion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Group_without_torsion&oldid=31930
This article was adapted from an original article by V.M. Kopytov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article