[[Root system]]
# Weyl group

The **Weyl group** $\mathcal{W}$ of a [[root system]] $\Phi \subset \mathbb{E}$ is the group of isometries generated by the reflections $\sigma_{\alpha}$ for $\alpha \in \Phi$. #m/def/geo
It is thus a subgroup of the [[Automorphism|automorphism group]] $\Aut \Phi$,
and thus both of $\opn O(\mathbb{E})$ and [[Symmetric group|$\Phi!$]],
in particular it is a [[finite group]].

## Properties

1. [[Conjugation of a Weyl element]]

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