Weyl group

Definition

Definition with symbols

Let $H\leq G$ 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 $\operatorname {Aut} (H)$ 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).

Weyl groups in algebraic groups

Further information: Weyl group of a maximal torus in a linear algebraic group

In the context of a linear algebraic group, the term Weyl group is typically used to refer to the Weyl group of a maximal torus in the group. If the linear algebraic group is over an algebraically closed field, then the maximal tori are all conjugate, and the Weyl groups are thus all isomorphic. Further, in this case, the normalizer of a maximal torus is actually an internal semidirect product of the maximal torus with another subgroup, and we can treat any of the possible complements as Weyl subgroups of the whole group.