Abstract
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 Sn, 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 Sn 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.