Torsion subgroup

For any abelian group ${\displaystyle (A,+)}$ and any prime number p the set A_Tp 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:

${\displaystyle A_{T_{p}}=\{a\in A\;|\;\exists n\in \mathbb {N} \;,p^{n}a=0\}.\;}$

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

${\displaystyle A_{T}\cong \bigoplus _{p\in P}A_{T_{p}}.\;}$

When A is a finite abelian group, A_Tp 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.

The 4-torsion subgroup of the quotient group of the complex numbers under addition by a lattice.

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

⟨ x, y | x² = y² = 1 ⟩

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.[2]
• 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.
• 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 ⊗ A to B ⊗ A is injective.
• Tensoring an abelian group A with Q (or any divisible group) kills torsion. That is, if T is a torsion group then T ⊗ Q = 0. For a general abelian group A with torsion subgroup T one has A ⊗ Q ≅ A/T ⊗ 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.

• Torsion (algebra)
• Torsion-free abelian group
• Torsion abelian group

• Epstein, David B. A.; Cannon, James W.; Holt, Derek F.; Levy, Silvio V. F.; Paterson, Michael S.; Thurston, William P. (1992), Word Processing in Groups, Boston, MA: Jones and Bartlett Publishers, ISBN 0-86720-244-0