# Paranormal implies weakly closed in intermediate nilpotent

## Statement

### Statement with symbols

Suppose $H \le K \le G$ are groups such that:

• $H$ is a Paranormal subgroup (?) of $G$.
• $K$ is a Nilpotent group (?).

Then, $H$ is a Weakly closed subgroup (?) of $K$.

## Definitions used

For these definitions, $H^g = g^{-1}Hg$ denotes the conjugate subgroup by $g \in G$. (This is the right-action convention; however, adopting a left-action convention does not alter any of the proof details).

### Paranormal subgroup

Further information: Paranormal subgroup

A subgroup $H$ of a group $G$ is termed paranormal in $G$ if for any $g \in G$, $H$ is a contranormal subgroup of $\langle H, H^g \rangle$; in other words, the normal closure of $H$ in $\langle H, H^g \rangle$ is the whole of $\langle H, H^g \rangle$.

### Weakly closed subgroup

Further information: Weakly closed subgroup

Suppose $H \le K \le G$ are groups. We say $H$ is weakly closed in $K$ with respect to $G$ if, for any $g \in G$ such that $H^g \le K$, we have $H^g \le H$.

## Proof

The proof follows directly from facts (1) and (2).