Coprime automorphism-invariant normal subgroup of Hall subgroup is normalizer-relatively normal

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Definition

Suppose G is a finite group, K is a Hall subgroup of G, and H is a Coprime automorphism-invariant normal subgroup (?) of G. Then, H is normal in the normalizer N_G(K). In other words, H is normalizer-relatively normal in K relative to G.

Related facts

Applications

Proof

Given: A finite group G, a Hall subgroup K of G, a coprime automorphism-invariant normal subgroup H of K.

To prove: H is normal in N_G(K).

Proof: Let g \in N_G(K). Suppose g has order m.

  1. We can write g = ab where a \in K and the order of b is relatively prime to the order of K.
  2. Conjugation by g can be expressed as a composite of conjugation by a and conjugation by b, and H is invariant under both: H is invariant under conjugation by b because this is an automorphism of coprime order. H is invariant under conjugation by a because H is normal in K and a is in K.