Torsion subgroup
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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 C2; 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 Z-module, 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 p-torsion subgroup, rather than p-power 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.