Editing Torsion subgroup (section)

==''p''-power torsion subgroups==
For any abelian group <math>(A, +)</math> and any [[prime number]] ''p'' the set ''A<sub>Tp</sub>'' 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''':

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

The torsion subgroup ''A<sub>T</sub>'' is isomorphic to the direct sum of its ''p''-power torsion subgroups over all prime numbers ''p'':

:<math>A_T \cong \bigoplus_{p\in P} A_{T_p}.\;</math>

When ''A'' is a finite abelian group, ''A<sub>Tp</sub>'' coincides with the unique [[Sylow subgroup|Sylow ''p''-subgroup]] of ''A''.

Each ''p''-power torsion subgroup of ''A'' is a [[characteristic subgroup|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.