Weakly normal implies weakly closed in intermediate nilpotent

Statement

Statement with symbols

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

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

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

Related facts

• Weakly normal implies intermediately subnormal-to-normal

Corollaries

• Paranormal implies weakly closed in intermediate nilpotent
• Pronormal implies weakly closed in intermediate nilpotent

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: Weakly normal subgroup

A subgroup $H$ of a group $G$ is termed paranormal in $G$ if for any $g\in G$, $H^{g}\leq N_{G}(H)$ implies $H^{g}\leq H$. In other words, it is weakly closed in its normalizer.

Weakly closed subgroup

Further information: Weakly closed subgroup

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

Facts used

1. Weakly normal implies intermediately subnormal-to-normal
2. Nilpotent implies every subgroup is subnormal

Proof

Given: $H\leq K\leq G$ with $H$ a paranormal subgroup of $G$ and $K$ a nilpotent group.

To prove: For any $g\in G$ such that $H^{g}\leq K$, we have $H^{g}\leq H$.

Proof: By fact (2), $H$ is a subnormal subgroup of $K$. By fact (1), $H$ is therefore normal in $K$. Thus, $K\leq N_{G}(H)$.

Since $H$ is weakly closed in $N_{G}(H)$ by definition, whenever $H^{g}\leq N_{G}(H)$, we get $H^{g}\leq H$. In particular, whenever $H^{g}\leq K$, we get $H^{g}\leq H$, so $H$ is weakly closed in $K$.