a short exact sequence of groupspage 8 where is an abelian variety, is of multiplicative type (meaning, is, geometrically, a product of tori and algebraic groups of the form ) and is a unipotent group. Characteristic p When the characteristic of the base field is p there is an analogous statement for an algebraic group : there exists a smallest subgroup such that is a unipotent group is an extension of an abelian variety by a group of multiplicative type. is unique up to commensurability in and is unique up to isogeny. Jordan decomposition Any element g of a linear algebraic group over a perfect field can be written uniquely as the product g = gu&hairsp;&hairsp;gs of commuting unipotent and semisimple elements gu and gs. In the case of the group GLn(C), this essentially says that any invertible complex matrix is conjugate to the product of a diagonal matrix and an upper triangular one, which is (more or less) the multiplicative version of the Jordan–Chevalley decomposition. There is also a version of the Jordan decomposition for groups: any commutative linear algebraic group over a perfect field is the product of a unipotent group and a semisimple group. See also Reductive group Unipotent representation Deligne–Lusztig theory References A. Borel, Linear algebraic groups, Ring theory Matrix theory Algebraic groups