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 by a normal subgroup
is a torsion-free group if and only if
implies
for all
and for any natural number
. An extension of a torsion-free group by a torsion-free group is a torsion-free group. If a group is residually a finite
-group (cf. Residually-finite group;
-group) with respect to two different prime numbers
, then it is a torsion-free group.
References
[1] | A.G. Kurosh, "The theory of groups" , 1–2 , Chelsea (1955–1956) (Translated from Russian) |
Group without torsion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Group_without_torsion&oldid=31930