On the Affine Weyl group of type

Source abstract

We study in this paper the affine Weyl group of type , [1]. Coxeter [1] showed that this group is infinite. We see in Bourbaki [2] that is a split extension of S n , the symmetric group of degree n , by a group of translations and of lattice of weights. is one of the crystallographic Coxeter groups considered by Maxwell [3], [4]. We prove the following: THEOREM 1. is a split extension of S n by the direct product of ( n − 1) copies of Z . THEOREM 2. The group is soluble of derived length 3, is soluble of derived length 4. For n > 4, the second derived group coincides with the first and so is not soluble for n > 4. THEOREM 3. The center of is trivial for n ≥ 3.