Weyl group (reviewed)

The Weyl group associated to a subgroup $H \subseteq G$ is the quotient $W = N_G(H) / Z_G(H)$ of the normalizer of $H$ by the centralizer of $H$. The terminology is taken from the theory of Lie groups, where there is a unique conjugacy class of maximal tori in a Lie group $G$ and if $T$ is any such maximal torus then $Z_G(T) = T$ has finite index in $N_G(T)$. In that context we may thus speak of the Weyl group of $G$.

When $G$ is an abstract group, we must specify $H$ as well, and the Weyl group naturally embeds into the automorphism group $\Aut(H)$ and gives the subset of automorphisms of $H$ that are induced by conjugation within $G$.