Weyl group

Definition

Definition with symbols

Let $H\le G$ be groups. The Weyl group of $H$ with respect to $G$ can be defined in the following equivalent ways:

• It is the group of those automorphisms of $H$ which extend to inner automorphisms of $G$
• It is the quotient group $N_G(H)/C_G(H)$ where $N_G(H)$ is the normalizer of $H$ in $G$ and $C_G(H)$ is the centralizer of $H$ in $G$.
• it is the image of the natural homomorphism from $N_G(H)$ to $\mathrm{A}\mathrm{u}\mathrm{t}(H)$ that sends $g\in N_G(H)$ to the automorphism of $H$ given via conjugation by $g$.

Related notions

Relation with subgroup properties

The Weyl group always contains the inner automorphism group of $H$ and lies inside the automorphism group of $H$. This gives two extreme subgroup properties:

• Fully normalized subgroup is a subgroup whose Weyl group is the whole automorphism group
• Central factor of normalizer is a subgroup whose Weyl group is precisely the inner automorphism group

For self-centralizing Abelian subgroups

In the particular case where $H=C_G(H)$, the Weyl group of $H$ is simply $N_G(H)/H$. This situation is quite common in the case of linear groups, for instance: each torus (for instance, the subgroup of invertible diagonal matrices) is self-centralizing in the general linear group, and hence its Weyl group is simply the quotient of its normalizer, by itself (this turns out to be the symmetric group).