On the Affine Weyl group of type A ˜ n − 1

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