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- Diagonalizable_group abstract "In mathematics, an affine algebraic group is said to be diagonalizable if it is isomorphic to a subgroup of Dn, the group of diagonal matrices. A diagonalizable group defined over k is said to split over k or k-split if the isomorphism is defined over k. This coincides with the usual notion of split for an algebraic group. Every diagonalizable group splits over the separable closure ks of k. Any closed subgroup and image of diagonalizable groups are diagonalizable. The torsion subgroup of a diagonalizable group is dense.The category of diagonalizable groups defined over k is equivalent to the category of finitely generated abelian group with Gal(k/ks)-equivariant morphisms without p-torsion. This is an analog of Poincaré duality and motivated the terminology.A diagonalizable k-group is said to be anisotropic if it has no nontrivial k-valued character.The so-called \"rigidity\" states that the identity component of the centralizer of a diagonalizable group coincides with the identity component of the normalizer of the group. The fact plays a crucial role in the structure theory of solvable groups.A connected diagonalizable group is called an algebraic torus (which is not necessarily compact, in contrast to a complex torus). A k-torus is a torus defined over k. The centralizer of a maximal torus is called a Cartan subgroup.".
- Diagonalizable_group wikiPageID "26817254".
- Diagonalizable_group wikiPageLength "1723".
- Diagonalizable_group wikiPageOutDegree "12".
- Diagonalizable_group wikiPageRevisionID "536704945".
- Diagonalizable_group wikiPageWikiLink Algebraic_closure.
- Diagonalizable_group wikiPageWikiLink Algebraic_torus.
- Diagonalizable_group wikiPageWikiLink Cartan_subgroup.
- Diagonalizable_group wikiPageWikiLink Category:Algebraic_groups.
- Diagonalizable_group wikiPageWikiLink Complex_torus.
- Diagonalizable_group wikiPageWikiLink Group_isomorphism.
- Diagonalizable_group wikiPageWikiLink Linear_algebraic_group.
- Diagonalizable_group wikiPageWikiLink Mathematics.
- Diagonalizable_group wikiPageWikiLink Poincaré_duality.
- Diagonalizable_group wikiPageWikiLink Rigidity_theorem_on_an_abelian_variety.
- Diagonalizable_group wikiPageWikiLink Splitting_lemma.
- Diagonalizable_group wikiPageWikiLink Torsion_subgroup.
- Diagonalizable_group wikiPageWikiLinkText "Diagonalizable group".
- Diagonalizable_group wikiPageWikiLinkText "diagonalizable group".
- Diagonalizable_group wikiPageUsesTemplate Template:Abstract-algebra-stub.
- Diagonalizable_group wikiPageUsesTemplate Template:Reflist.
- Diagonalizable_group subject Category:Algebraic_groups.
- Diagonalizable_group type Group.
- Diagonalizable_group type Group.
- Diagonalizable_group type Variety.
- Diagonalizable_group comment "In mathematics, an affine algebraic group is said to be diagonalizable if it is isomorphic to a subgroup of Dn, the group of diagonal matrices. A diagonalizable group defined over k is said to split over k or k-split if the isomorphism is defined over k. This coincides with the usual notion of split for an algebraic group. Every diagonalizable group splits over the separable closure ks of k. Any closed subgroup and image of diagonalizable groups are diagonalizable.".
- Diagonalizable_group label "Diagonalizable group".
- Diagonalizable_group sameAs Q5270375.
- Diagonalizable_group sameAs m.0bmg_f2.
- Diagonalizable_group sameAs Q5270375.
- Diagonalizable_group wasDerivedFrom Diagonalizable_group?oldid=536704945.
- Diagonalizable_group isPrimaryTopicOf Diagonalizable_group.