Weyl group

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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 \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:

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).