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| 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]]. An abelian group ''A'' is called a '''torsion''' (or '''[[periodic group|periodic]]''') group if every element of ''A'' has finite order and is called '''torsion-free''' if every element of ''A'' except the [[identity element|identity]] is of infinite order.
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| The proof that ''A<sub>T</sub>'' is closed under addition relies on the commutativity of addition (see examples section).
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| 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 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).
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| If ''A'' is 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.
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| ==''p''-power torsion subgroups==
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| 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''':
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| :<math>A_{T_p}=\{g\in A \;|\; \exists n\in \mathbb{N}\;, p^n g = 0\}.\;</math>
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| The torsion subgroup ''A<sub>T</sub>'' is isomorphic to the direct sum of its ''p''-power torsion subgroups over all prime numbers ''p'':
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| :<math>A_T \cong \bigoplus_{p\in P} A_{T_p}.\;</math>
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| When ''A'' is a finite abelian group, ''A<sub>Tp</sub>'' coincides with the unique [[Sylow subgroup|Sylow p-subgroup]] of ''A''.
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| 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.
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| 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.
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| ==Examples and further results==
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| [[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. ]]
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| *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]]:
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| : ⟨ ''x'', ''y'' | ''x''² = ''y''² = 1 ⟩
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| :the element ''xy'' is a product of two torsion elements, but has infinite order.
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| * The torsion elements in a [[nilpotent group]] form a normal subgroup.<ref>See Epstein & Cannon (1992) [http://books.google.com/books?id=DQ84QlTr-EgC&pg=PA167 p. 167]</ref>
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| *Obviously, every finite abelian group is a torsion group. Not every torsion group is finite however: consider the [[direct sum of groups|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.
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| *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'''.
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| *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]].
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| *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]].
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| *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'''.
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| ==See also==
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| * [[Torsion (algebra)]]
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| * [[Torsion-free abelian group]]
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| * Epstein, D. B. A., Cannon, James W.. ''Word processing in groups''. A K Peters, 1992. ISBN 0-86720-244-0
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| {{DEFAULTSORT:Torsion Subgroup}}
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| [[Category:Abelian group theory]]
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| [[de:Torsion (Algebra)]]
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ңеllo, I'm Kristen, a 26 yeɑr old from Poznan, Poland.
My hobbies include (but are not limited to) Nordic skating, Fishing аnd watching Supernatural.
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