Coprime automorphism-invariant normal subgroup of Hall subgroup is normalizer-relatively normal
Suppose is a finite group, is a Hall subgroup of , and is a Coprime automorphism-invariant normal subgroup (?) of . Then, is normal in the normalizer . In other words, is normalizer-relatively normal in relative to .
Given: A finite group , a Hall subgroup of , a coprime automorphism-invariant normal subgroup of .
To prove: is normal in .
Proof: Let . Suppose has order .
- We can write where and the order of is relatively prime to the order of .
- Conjugation by can be expressed as a composite of conjugation by and conjugation by , and is invariant under both: is invariant under conjugation by because this is an automorphism of coprime order. is invariant under conjugation by because is normal in and is in .