Torsion subgroup

$p$-power torsion subgroups

For any abelian group (A, +) and any prime number p the set A_{T_p} of elements of A that have order a power of p is a subgroup called the ** p-power torsion subgroup** or, more loosely, the ** p-torsion subgroup**:

:A_{T_p}={a\in A ;|; \exists n\in \mathbb{N};, p^n a = 0}.;

The torsion subgroup A_T is isomorphic to the direct sum of its p-power torsion subgroups over all prime numbers p:

:A_T \cong \bigoplus_{p\in P} A_{T_p}.;

When A is a finite abelian group, A_{T_p} coincides with the unique Sylow p-subgroup of A.

Each p-power torsion subgroup of A is a fully characteristic subgroup. More strongly, any homomorphism between abelian groups sends each p-power torsion subgroup into the corresponding p-power torsion subgroup.

For each prime number p, this provides a functor from the category of abelian groups to the category of p-power torsion groups that sends every group to its p-power torsion subgroup, and restricts every homomorphism to the p-torsion subgroups. The product over the set of all prime numbers of the restriction of these functors to the category of torsion groups, is a faithful functor from the category of torsion groups to the product over all prime numbers of the categories of p-torsion groups. In a sense, this means that studying p-torsion groups in isolation tells us everything about torsion groups in general.

Examples and further results

• The torsion subset of a non-abelian group is not, in general, a subgroup. For example, in the infinite dihedral group, which has presentation:

: \langle x,y \mid x^2=y^2=1 \rangle

:the element xy is a product of two torsion elements, but has infinite order.

• The torsion elements in a nilpotent group form a normal subgroup.
• 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 \mathbb{Q}/\mathbb{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 \mathbb{Q}.
• 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.
• 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\otimes A\to B\otimes A is injective.
• Tensoring an abelian group A with \mathbb{Q} (or any divisible group) kills torsion. That is, if T is a torsion group then T\otimes\mathbb{Q}=0. For a general abelian group A with torsion subgroup T one has A\otimes\mathbb{Q}\cong A/T\otimes\mathbb{Q}.
• Taking the torsion subgroup makes torsion abelian groups into a coreflective subcategory of abelian groups, while taking the quotient by the torsion subgroup makes torsion-free abelian groups into a reflective subcategory.

Notes

References

• {{citation

de:Torsion (Algebra)

References

1. See Epstein & Cannon (1992) [https://books.google.com/books?id=DQ84QlTr-EgC&pg=PA167 p. 167]