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 <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.

====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>