Weakly normal implies weakly closed in intermediate nilpotent
Statement with symbols
Suppose are groups such that:
Then, is a Weakly closed subgroup (?) of .
- Paranormal implies weakly closed in intermediate nilpotent
- Pronormal implies weakly closed in intermediate nilpotent
For these definitions, denotes the conjugate subgroup by . (This is the right-action convention; however, adopting a left-action convention does not alter any of the proof details).
Further information: Weakly normal subgroup
A subgroup of a group is termed paranormal in if for any , implies . In other words, it is weakly closed in its normalizer.
Weakly closed subgroup
Further information: Weakly closed subgroup
Suppose are groups. We say is weakly closed in with respect to if, for any such that , we have .
- Weakly normal implies intermediately subnormal-to-normal
- Nilpotent implies every subgroup is subnormal
Given: with a paranormal subgroup of and a nilpotent group.
To prove: For any such that , we have .
Proof: By fact (2), is a subnormal subgroup of . By fact (1), is therefore normal in . Thus, .
Since is weakly closed in by definition, whenever , we get . In particular, whenever , we get , so is weakly closed in .