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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 wikiPageLength "6075".
- Torsion_subgroup wikiPageOutDegree "38".
- Torsion_subgroup wikiPageRevisionID "663694223".
- Torsion_subgroup wikiPageWikiLink Abelian_group.
- Torsion_subgroup wikiPageWikiLink Category:Abelian_group_theory.
- Torsion_subgroup wikiPageWikiLink Category_of_abelian_groups.
- Torsion_subgroup wikiPageWikiLink Characteristic_subgroup.
- Torsion_subgroup wikiPageWikiLink Coreflective_subcategory.
- Torsion_subgroup wikiPageWikiLink Countable_set.
- Torsion_subgroup wikiPageWikiLink Countably_infinite.
- Torsion_subgroup wikiPageWikiLink Covariant_functor.
- Torsion_subgroup wikiPageWikiLink Cyclic_group.
- Torsion_subgroup wikiPageWikiLink Direct_sum_of_groups.
- Torsion_subgroup wikiPageWikiLink Divisible_group.
- Torsion_subgroup wikiPageWikiLink Factor_group.
- Torsion_subgroup wikiPageWikiLink Faithful_functor.
- Torsion_subgroup wikiPageWikiLink Finitely_generated_abelian_group.
- Torsion_subgroup wikiPageWikiLink Flat_module.
- Torsion_subgroup wikiPageWikiLink Free_abelian_group.
- Torsion_subgroup wikiPageWikiLink Full_and_faithful_functors.
- Torsion_subgroup wikiPageWikiLink Functor.
- Torsion_subgroup wikiPageWikiLink Generating_set_of_a_group.
- Torsion_subgroup wikiPageWikiLink Identity_element.
- Torsion_subgroup wikiPageWikiLink If_and_only_if.
- Torsion_subgroup wikiPageWikiLink Infinite_dihedral_group.
- Torsion_subgroup wikiPageWikiLink Injective.
- Torsion_subgroup wikiPageWikiLink Injective_function.
- Torsion_subgroup wikiPageWikiLink Module_(mathematics).
- Torsion_subgroup wikiPageWikiLink Nilpotent_group.
- Torsion_subgroup wikiPageWikiLink Order_(group_theory).
- Torsion_subgroup wikiPageWikiLink Periodic_group.
- Torsion_subgroup wikiPageWikiLink Presentation_of_a_group.
- Torsion_subgroup wikiPageWikiLink Quotient_group.
- Torsion_subgroup wikiPageWikiLink Rank_of_an_abelian_group.
- Torsion_subgroup wikiPageWikiLink Rational_number.
- Torsion_subgroup wikiPageWikiLink Reflective_subcategory.
- Torsion_subgroup wikiPageWikiLink Subgroup.
- Torsion_subgroup wikiPageWikiLink Sylow_subgroup.
- Torsion_subgroup wikiPageWikiLink Sylow_theorems.
- Torsion_subgroup wikiPageWikiLink Tensor_product_of_abelian_groups.
- Torsion_subgroup wikiPageWikiLink Tensor_product_of_modules.
- Torsion_subgroup wikiPageWikiLink Torsion-free_abelian_group.
- Torsion_subgroup wikiPageWikiLink Torsion_(algebra).
- Torsion_subgroup wikiPageWikiLink Torsion_abelian_group.
- Torsion_subgroup wikiPageWikiLink Torsion_group.
- Torsion_subgroup wikiPageWikiLink File:Lattice_torsion_points.svg.
- Torsion_subgroup wikiPageWikiLinkText "Torsion subgroup".
- Torsion_subgroup wikiPageWikiLinkText "torsion points".
- Torsion_subgroup wikiPageWikiLinkText "torsion subgroup".
- Torsion_subgroup wikiPageWikiLinkText "torsion".
- Torsion_subgroup wikiPageWikiLinkText "torsionfree".
- Torsion_subgroup hasPhotoCollection Torsion_subgroup.
- Torsion_subgroup wikiPageUsesTemplate Template:Reflist.
- Torsion_subgroup subject Category:Abelian_group_theory.
- Torsion_subgroup hypernym Subgroup.
- Torsion_subgroup type EthnicGroup.
- 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 "Torsion subgroup".
- Torsion_subgroup sameAs Sottogruppo_di_torsione.
- Torsion_subgroup sameAs 捩れ部分群.
- Torsion_subgroup sameAs 꼬임_부분군.
- Torsion_subgroup sameAs Podgrupa_torsyjna.
- Torsion_subgroup sameAs Subgrupo_de_torsão.
- Torsion_subgroup sameAs m.01274g.
- Torsion_subgroup sameAs Подгруппа_кручения.
- Torsion_subgroup sameAs Підгрупа_кручення.
- Torsion_subgroup sameAs Q2293759.
- Torsion_subgroup sameAs Q2293759.
- Torsion_subgroup sameAs 撓子群.
- Torsion_subgroup wasDerivedFrom Torsion_subgroup?oldid=663694223.
- Torsion_subgroup depiction Lattice_torsion_points.svg.
- Torsion_subgroup isPrimaryTopicOf Torsion_subgroup.