# Group without torsion

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://www.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