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- Torsion_subgroup abstract "In the theory of abelian groups, the torsion subgroup AT of an abelian group A is the subgroup of A consisting of all elements that have finite order. An abelian group A is called a torsion (or periodic) group if every element of A has finite order and is called torsion-free if every element of A except the identity is of infinite order. The proof that AT is closed under addition relies on the commutativity of addition (see examples section).If A is abelian, then the torsion subgroup T is a 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).If A is finitely generated and abelian, then it can be written as the 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 groups.".
- Torsion_subgroup thumbnail Lattice_torsion_points.svg?width=300.
- Torsion_subgroup wikiPageID "144052".
- Torsion_subgroup wikiPageRevisionID "572502674".
- Torsion_subgroup hasPhotoCollection Torsion_subgroup.
- Torsion_subgroup subject Category:Abelian_group_theory.
- Torsion_subgroup comment "In the theory of abelian groups, the torsion subgroup AT of an abelian group A is the subgroup of A consisting of all elements that have finite order. An abelian group A is called a torsion (or periodic) group if every element of A has finite order and is called torsion-free if every element of A except the identity is of infinite order.".
- Torsion_subgroup label "Podgrupa torsyjna".
- Torsion_subgroup label "Sottogruppo di torsione".
- Torsion_subgroup label "Subgrupo de torsão".
- Torsion_subgroup label "Torsion subgroup".
- Torsion_subgroup label "Подгруппа кручения".
- Torsion_subgroup label "撓子群".
- Torsion_subgroup sameAs Sottogruppo_di_torsione.
- Torsion_subgroup sameAs 꼬임_부분군.
- Torsion_subgroup sameAs Podgrupa_torsyjna.
- Torsion_subgroup sameAs Subgrupo_de_torsão.
- Torsion_subgroup sameAs m.01274g.
- Torsion_subgroup sameAs Q2293759.
- Torsion_subgroup sameAs Q2293759.
- Torsion_subgroup wasDerivedFrom Torsion_subgroup?oldid=572502674.
- Torsion_subgroup depiction Lattice_torsion_points.svg.
- Torsion_subgroup isPrimaryTopicOf Torsion_subgroup.
