Pronormal implies intermediately subnormal-to-normal

Statement
A pronormal subgroup of a group is intermediately subnormal-to-normal: it is normal in any intermediate subgroup in which it is subnormal.

Intermediate properties

 * Weakly pronormal subgroup
 * Paranormal subgroup
 * Join of pronormal subgroups
 * Polynormal subgroup
 * Weakly normal subgroup

Facts used

 * 1) uses::Pronormal and subnormal implies normal
 * 2) uses::Pronormality satisfies intermediate subgroup condition: A pronormal subgroup is also pronormal in every intermediate subgroup.

Proof
Given: $$H \le K \le G$$, such that $$H$$ is pronormal in $$G$$ and subnormal in $$K$$.

To prove: $$H$$ is normal in $$K$$.

Proof:


 * 1) $$H$$ is pronormal in $$K$$: This follows from fact (2) and the given datum that $$H$$ is pronormal in $$G$$.
 * 2) $$H$$ is normal in $$K$$: This follows from fact (1), the previous step, and the given datum that $$H$$ is subnormal in $$K$$.