# Weyl group

From Groupprops

## Contents

## Definition

### Definition with symbols

Let be groups. The **Weyl group** of with respect to can be defined in the following equivalent ways:

- It is the group of those automorphisms of which extend to inner automorphisms of
- It is the quotient group where is the normalizer of in and is the centralizer of in .
- it is the image of the natural homomorphism from to that sends to the automorphism of given via conjugation by .

## Related notions

### Relation with subgroup properties

The Weyl group always contains the inner automorphism group of and lies inside the automorphism group of . 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 , the Weyl group of is simply . 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).