Pronormal implies intermediately subnormal-to-normal

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., pronormal subgroup) must also satisfy the second subgroup property (i.e., intermediately subnormal-to-normal subgroup)
View all subgroup property implications | View all subgroup property non-implications
Get more facts about pronormal subgroup|Get more facts about intermediately subnormal-to-normal subgroup

Statement

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

Related properties

Intermediate properties

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

Facts used

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

Proof

Given: $H\leq K\leq G$, 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 $K$.