Nilpotent join of pronormal subgroups is pronormal
- and normalize each other.
- is also a pronormal subgroup of .
- Pronormality satisfies intermediate subgroup condition
- Nilpotent implies every subgroup is subnormal
- Pronormal and subnormal implies normal
- Pronormality is normalizing join-closed
Given: are pronormal subgroups, and is nilpotent.
To prove: and normalize each other, and is also pronormal.
Proof: By fact (1), are both pronormal in . By fact (2), they are also both subnormal in . By fact (3), are both normal in . Thus, we have , and that and normalize each other. By fact (4), we obtain that is also pronormal.