Weyl group

The Weyl group $W$ of a root system $Φ \subset 𝔼$ is the group of isometries generated by the reflections $𝜎 𝛼$ for $𝛼 \in Φ$ . geo It is thus a subgroup of the automorphism group $A u t Φ$ , and thus both of $O (𝔼)$ and [[Symmetric group| $Φ!$ ]], in particular it is a finite group.