View source for 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 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====
<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>