In this paper we study the Weyl groups of reduced extended affine root syst
ems, the root systems of extended affine Lie algebras. We start by describi
ng the extended affine Weyl group as a semidirect product of a finite Wcyl
group and a Heisenberg-like normal subgroup. This provides a unique express
ion for the Weyl group elements (in terms of some naturally arisen transfor
mations) which is crucial in the further study of extended affine Weyl grou
ps. We use this to give a presentation, called a presentation by conjugatio
n, for an important subclass of extended affine Weyl groups. Using a new no
tion, called the index which is an invariant of the extended affine root sy
stems, we show that one of the important features of finite and affine root
systems (related to Weyl group) holds for the class of extended affine roo
t systems. (C) 1999 Academic Press.