In the context of group theory, particularly in the study of abelian groups (and more generally, in the context of modules over a ring), the **torsion subgroup** is an important concept. The torsion subgroup of an abelian group \( G \) is defined as the set of elements in \( G \) that have finite order.
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The torsion subgroup of an abelian group consists of all elements of finite order. For a finitely generated abelian group, it is the finite direct summand complementary to the free abelian part.