Difference between revisions of "Group without torsion"
From Encyclopedia of Mathematics
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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 | + | 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=18602
Group without torsion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Group_without_torsion&oldid=18602
This article was adapted from an original article by V.M. Kopytov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article