A torsion-free group is a group whose only element of finite order is its identity
element. Thus 
for a positive integer 
implies 
. Equivalently, its group torsion
is trivial. The infinite cyclic
group 
and all free groups are examples, whereas a nontrivial
finite group is not torsion-free (Brown 1982).
The condition eliminates finite stabilizers in a proper group action. It also enters the Kadison-Kaplansky conjecture for projections in reduced group C*-algebras.
