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In the theory of algebraic groups, a **Cartan subgroup** of a connected linear algebraic group over a (not necessarily algebraically closed) field is the centralizer of a maximal torus. Cartan subgroups are smooth (equivalently reduced), connected and nilpotent. If is algebraically closed, they are all conjugate to each other. ^{[1]}

Notice that in the context of algebraic groups a *torus* is an algebraic group
such that the base extension (where is the algebraic closure of ) 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 is reductive (in particular, if it is semi-simple), then a torus is maximal if and only if it is its own centraliser ^{[2]} and thus Cartan subgroups of are precisely the maximal tori.

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.

**^**Milne (2017), Proposition 17.44.**^**Milne (2017), Corollary 17.84.

- Borel, Armand (1991-12-31).
*Linear algebraic groups*. ISBN 3-540-97370-2. - Lang, Serge (2002).
*Algebra*. Springer. ISBN 978-0-387-95385-4. - Milne, J. S. (2017),
*Algebraic Groups: The Theory of Group Schemes of Finite Type over a Field*, Cambridge University Press, doi:10.1017/9781316711736, ISBN 978-1107167483, MR 3729270 - Popov, V. L. (2001) [1994], "Cartan subgroup",
*Encyclopedia of Mathematics*, EMS Press - Springer, Tonny A. (1998),
*Linear algebraic groups*, Progress in Mathematics, vol. 9 (2nd ed.), Boston, MA: Birkhäuser Boston, ISBN 978-0-8176-4021-7, MR 1642713