In the theory of algebraic groups, a Cartan subgroup of a connected linear algebraic group G over a (not necessarily algebraically closed) field k is the centralizer of a maximal torus. Cartan subgroups are smooth (equivalently reduced), connected and nilpotent. If k is algebraically closed, they are all conjugate to each other. Notice that in the context of algebraic groups a torus is an algebraic group T such that the base extension (where \bar{k} is the algebraic closure of k) is isomorphic to the product of a finite number of copies of the. Maximal such subgroups have in the theory of algebraic groups a role that is similar to that of maximal tori in the theory of Lie groups. If G is reductive (in particular, if it is semi-simple), then a torus is maximal if and only if it is its own centraliser and thus Cartan subgroups of G are precisely the maximal tori.

Example

The general linear groups are reductive. The diagonal subgroup is clearly a torus (indeed a split torus, since it is product of n copies of already before any base extension), and it can be shown to be maximal. Since is reductive, the diagonal subgroup is a Cartan subgroup.