Pronormal implies weakly closed in intermediate nilpotent
Contents
Statement
Statement with symbols
Suppose are groups such that:
- is a Pronormal subgroup (?) of .
- is a Nilpotent group (?).
Then, is a Weakly closed subgroup (?) of .
Related facts
Stronger facts
- Paranormal implies weakly closed in intermediate nilpotent: Paranormality is a somewhat weaker assumption than pronormality, and hence the corresponding result is stronger. The proof is almost the same.
Definitions used
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).
Pronormal subgroup
Further information: Pronormal subgroup
A subgroup of a group is termed paranormal in if for any , there exists such that .
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 .
Facts used
- Pronormal implies intermediately subnormal-to-normal
- Nilpotent implies every subgroup is subnormal
- Normality satisfies intermediate subgroup condition
Proof
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 .
Now suppose is such that . Then, . By fact (3), is normal in . Thus, for any , we have .
By the definition of pronormality, we also have such that . Since , we get , completing the proof.
References
Textbook references
- Finite Groups by Daniel Gorenstein, ISBN 0821843427, ^{More info}, Page 266, Exercise 9, Chapter 7