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 .
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).