About: Diagonalizable group

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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 a field 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.

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rdf:type	yago:Abstraction100002137 yago:Group100031264 yago:WikicatAlgebraicGroups
rdfs:label	Diagonalizable group (en)
rdfs: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 a field 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. (en)
dct:subject	Algebraic groups
Wikipage page ID	26817254 (xsd:integer)
Wikipage revision ID	1093120390 (xsd:integer)
Link from a Wikipage to another Wikipage	Algebraic torus Algebraic groups Mathematics Equivalence of categories Complex torus Subgroup Poincaré duality Field (mathematics) Diagonal subgroup Isomorphic Characteristic (algebra) Diagonal matrices Splitting lemma Cartan subgroup Category (mathematics) Separable closure Solvable group Finitely generated abelian group Torsion subgroup Affine algebraic group Normalizer Centralizer
sameAs	Diagonalizable group Diagonalizable group Diagonalizable group Diagonalizable group
dbp:wikiPageUsesTemplate	dbt:Reflist dbt:Abstract-algebra-stub
has 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 a field 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 groups with Gal(ks/k)-equivariant morphisms without p-torsion, if k is of characteristic p. 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. (en)
prov:wasDerivedFrom	wikipedia-en:Diagonalizable_group?oldid=1093120390&ns=0
page length (characters) of wiki page	1880 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Diagonalizable_group
is Link from a Wikipage to another Wikipage of	Diagonal subgroup Diagonalizable matrix Group scheme
is foaf:primaryTopic of	wikipedia-en:Diagonalizable_group

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