International Journal of Mathematics and Mathematical Sciences
Volume 10 (1987), Issue 1, Pages 147-154
On the Affine Weyl group of type
Department of Mathematical Sciences, University of Petroleum and Minerals, Dhahran, Saudi Arabia
Received 4 April 1985; Revised 26 March 1986
Copyright © 1987 Muhammad A. Albar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We study in this paper the affine Weyl group of type , . Coxeter  showed that this group is infinite. We see in Bourbaki  that is a split extension of , the symmetric group of degree , by a group of translations and of lattice of weights. is one of the crystallographic Coxeter groups considered by Maxwell , .
We prove the following:
THEOREM 1. is a split extension of by the direct product of copies of .
THEOREM 2. The group is soluble of derived length , is soluble of derived length . For , the second derived group coincides with the first and so is not soluble for .
THEOREM 3. The center of is trivial for .