Torsion subgroup

In group theory, the torsion subgroup of an abelian group G is the subgroup T of G which consists of all elements of G which have finite order. A group G is called torsion free if every element of G except the identity is of infinite order, and torsion (or periodic) if every element of G has finite order. Of course every finite abelian group is a torsion group. If G is abelian, then the torsion subgroup T is a characteristic subgroup of G (even fully characteristic) and the factor group G/T is torsion free.

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 G additively. Suppose 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 G is not abelian, and indeed in this case the set of all elements of G 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

A torsion group need not be finite; for example the direct sum of a countable number of copies of the cyclic group C₂ is a torsion group, every element of which 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 G is finitely generated and abelian, then it can be written as the direct sum of its torsion subgroup and a torsion free subgroup. In any decomposition of G 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 G 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 G and H are abelian groups with torsion subgroups T(G) and T(H), respectively, and f : G → H is a group homomorphism, then f(T(G)) is a subset of T(H). 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 G is torsion free if and only if it is flat as a Z-module, which means that whenever K is a subgroup of the abelian group H, then the natural map between the tensor products K ⊗ G and H ⊗ G is injective.