# 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