Torsion subgroup

In group theory, the torsion subgroup of an abelian group A is the subgroup of A consisting of all elements that have finite order. The abelian group A is called torsion free if every element of A except the identity is of infinite order, and torsion (or periodic) if every element of A has finite order.
Proof of the subgroup property
The set T of all elements of finite order in an abelian group indeed forms a subgroup: write the group A additively, and let mx denote the sum of m copies of x, e.g. 3x = x + x + x. The identity element 0 has order 1 and is therefore in T. If x and y are in T and m is the product of their orders, then m (x + y) = mx + my = 0 + 0 = 0, and so x + y is in T.
Note that this proof does not work if A is not abelian, and indeed in this case the set of all elements of A of finite order is not necessarily a subgroup.
Consider for example the infinite dihedral group, which has presentation ({x,y}, {x² = y² = 1}). This group is of countable infinite order, and in particular the element xy has infinite order. Since the group is generated by elements x and y which have order 2, the subset of finite elements generates the entire group.
Examples and further properties
Of course every finite abelian group is a torsion group. Not every torsion group is finite however: consider the direct sum of a countable number of copies of the cyclic group C_{2}; this is a torsion group since every element has order 2. Nor need there be an upper bound on the orders of elements in a torsion group if it isn't finitely generated, as the example of the factor group Q/Z shows.
Every free abelian group is torsion free, but the converse is not true, as is shown by the additive group of the rational numbers Q.
If A is abelian, then the torsion subgroup T is a characteristic subgroup of A (even fully characteristic) and the factor group A/T is torsion free.
If A is finitely generated and abelian, then it can be written as the direct sum of its torsion subgroup T and a torsion free subgroup. In any decomposition of A as a direct sum of a torsion subgroup S and a torsion free subgroup, S must equal T (but the torsion free subgroup is not uniquely determined). This is an important first step in the classification of finitely generated abelian groups.
Even if A is not finitely generated, the size of its torsion free part is uniquely determined, as is explained in more detail in the article on rank of an abelian group.
If A and B are abelian groups with torsion subgroups T(A) and T(B), respectively, and f : A → B is a group homomorphism, then f(T(A)) is a subset of T(B). We can thus define a functor T which assigns to each abelian group its torsion subgroup and to each homomorphism its restriction to the torsion subgroups.
An abelian group A is torsion free if and only if it is flat as a Zmodule, which means that whenever C is a subgroup of some abelian group B, then the natural map from the tensor product C ⊗ A to B ⊗ A is injective.
Prime power torsion
Within the torsion subgroup there is a subgroup associated to each prime number p, of elements that are killed by some power of p. This is often called the ptorsion subgroup, rather than ppower torsion subgroup which is more strictly accurate. In the case of a finite abelian group A it coincides with the Sylow subgroup for p, and A is up to isomorphism the direct sum of these subgroups.