The torsion subgroup of an Abelian group 
, written additively, is the subgroup
consisting of all elements 
such that 
for some positive integer 
. Equivalently, it consists of the elements
of finite order (The Stacks
Project Authors 2026). If 
and 
, then 
, which proves closure under subtraction. The torsion
subgroup is trivial iff 
is a torsion-free group.
For a general non-Abelian group, the elements of finite order need not form a subgroup. Thus the group torsion of an arbitrary group is not automatically a torsion subgroup.
