Weakly closed implies conjugation-invariantly relatively normal in finite group
Suppose are groups such that is a Weakly closed subgroup (?) of relative to . Then, is a Conjugation-invariantly relatively normal subgroup (?) of relative to , viz., is normal in every conjugate of in containing it.
Given: Groups such that is weakly closed in with respect to .
To prove: If is such that , then is normal in .
- : Since , we have . Now, since is weakly closed in , we get that .
- : Since is finite, and conjugation by is an automorphism, the sizes of and are the same. This, along with the previous step, yields .
- : This follows from the previous step by conjugating both sides by .
- is weakly closed in : Since is weakly closed in , and conjugation by is an automorphism, is weakly closed in . by the previos step, so is weakly closed in .
- is normal in : This follows from the previous step, and fact (1).