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.