Editing Torsion subgroup

{{Short description|Subgroup of an abelian group consisting of all elements of finite order}}
In the theory of [[abelian group]]s, the '''torsion subgroup''' ''A<sub>T</sub>'' of an abelian group ''A'' is the [[subgroup]] of ''A'' consisting of all elements that have finite [[order (group theory)|order]] (the '''torsion elements''' of ''A''<ref>{{citation | last=Serge|first = Lang  
|authorlink=Serge Lang
| title = Algebra
|pages=42 
|edition=3rd
| publisher = Addison-Wesley 
| year = 1993 
| isbn = 0-201-55540-9}}</ref>). An abelian group ''A'' is called a [[torsion group]] (or periodic group) if every element of ''A'' has finite order and is called [[Torsion-free abelian group|torsion-free]] if every element of ''A'' except the [[identity element|identity]] is of infinite order.

The proof that ''A<sub>T</sub>'' is closed under the group operation relies on the commutativity of the operation (see examples section).

If ''A'' is abelian, then the torsion subgroup ''T'' is a [[characteristic subgroup|fully characteristic subgroup]] of ''A'' and the factor group ''A''/''T'' is torsion-free. There is a [[covariant functor]] from the [[category of abelian groups]] to the category of torsion groups that sends every group to its torsion subgroup and every [[Group homomorphism|homomorphism]] to its restriction to the torsion subgroup. There is another covariant functor from the category of abelian groups to the category of torsion-free groups that sends every group to its quotient by its torsion subgroup, and sends every homomorphism to the obvious induced homomorphism (which is easily seen to be well-defined).

If ''A'' is [[finitely generated group|finitely generated]] and abelian, then it can be written as the [[direct sum of groups|direct sum]] of its torsion subgroup ''T'' and a torsion-free subgroup (but this is not true for all infinitely generated abelian groups). 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 a key step in the classification of [[finitely generated abelian group]]s.

==''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.

==Examples and further results==
[[Image:Lattice torsion points.svg|right|thumb|200px|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 of a group|presentation]]:

:  <math>\langle x,y \mid x^2=y^2=1 \rangle </math>

: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]].<ref>See Epstein & Cannon (1992) [https://books.google.com/books?id=DQ84QlTr-EgC&pg=PA167 p. 167]</ref>
*Every finite abelian group is a torsion group. Not every torsion group is finite however: consider the direct sum of a [[countably infinite|countable]] number of copies of the [[cyclic group]] ''C''<sub>2</sub>; 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 [[generating set of a group|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 number]]s '''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 module|flat]] as a '''Z'''-[[module (mathematics)|module]], which means that whenever ''C'' is a subgroup of some abelian group ''B'', then the natural map from the [[tensor product of abelian groups|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]].

==See also==
* [[Torsion (algebra)]]
* [[Torsion-free abelian group]]
* [[Torsion abelian group]]

==Notes==
{{Reflist}}

==References==
*{{citation
 | last1 = Epstein | first1 = David B. A. | authorlink1 = David B. A. Epstein
 | last2 = Cannon | first2 = James W. | authorlink2 = James W. Cannon
 | last3 = Holt | first3 = Derek F.
 | last4 = Levy | first4 = Silvio V. F.
 | last5 = Paterson | first5 = Michael S. | authorlink5 = Mike Paterson
 | last6 = Thurston | first6 = William P. | authorlink6 = William Thurston
 | title = Word Processing in Groups | title-link = Word Processing in Groups
 | publisher = Jones and Bartlett Publishers | location = Boston, MA | year = 1992 | isbn = 0-86720-244-0}}

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[[Category:Abelian group theory]]

[[de:Torsion (Algebra)]]