# Nilpotent join of pronormal subgroups is pronormal

From Groupprops

## Statement

Suppose are Pronormal subgroup (?)s, such that is a Nilpotent group (?). Then:

- and normalize each other.
- is also a pronormal subgroup of .

## Facts used

- Pronormality satisfies intermediate subgroup condition
- Nilpotent implies every subgroup is subnormal
- Pronormal and subnormal implies normal
- Pronormality is normalizing join-closed

## Proof

**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.