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I'm posting this question on behalf of someone without access to mathoverflow:

Can anybody give me a reliable reference (not a proof) to the following statement? Up to isomorphism, there are only finitely many Weyl groups of reductive algebraic groups of rank r.

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    $\begingroup$ There are at least three books called "Linear algebraic groups", one by Borel, one by Springer and one by Humphreys. The statement follows from the classification of reductive groups which is treated in each of these books. $\endgroup$
    – Matthias Wendt
    Commented Sep 19, 2014 at 12:06
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    $\begingroup$ I wonder how it's possible to have no access to mathoverflow... $\endgroup$
    – Tom De Medts
    Commented Sep 19, 2014 at 12:27

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A Weyl group is a crystallographic finite reflection group, i.e. it is a finite group generated by reflections in Euclidean space that preserves a lattice in the Euclidean space. So it suffices to prove that there are only finitely many crystallographic finite reflection groups of a given rank $r$. These are classified by Dynkin diagrams, for which there is a standard argument enumerating them, which shows that there are only finitely many with $r$ nodes. (In the classification scheme, the rank of the group is given by the number of nodes in the Dynkin diagram.)

This is very well known within the fields of Coxeter groups and Lie theory. The best reference for people outside these fields where this can be found is, in my opinion, Humphreys's book Finite Reflection Groups. The result you are looking for is stated and proved in Chapter 2 of this text. The result is also in Bourbaki's Lie Groups and Lie Algebras, Chapters IV - VI.

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