Pronormality is not transitive

This article gives the statement, and possibly proof, of a subgroup property (i.e., pronormal subgroup) not satisfying a subgroup metaproperty (i.e., transitive subgroup property).
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Statement

Verbal statement

A pronormal subgroup of a pronormal subgroup need not be pronormal in the whole group.

Statement with symbols

It is possible to have a group $G$ with subgroups $H\leq K\leq G$ such that $H$ is pronormal in $K$ and $K$ is pronormal in $G$, but $H$ is not pronormal in $G$.

Related facts

Generalization and other instances

• Subgroup property between normal and subnormal-to-normal is not transitive: The proof as given for pronormality generalizes to any subgroup property that is implied by normality, and such that any subnormal subgroup satisfying the property is normal.

Other instances of the generalization include:

• Paranormality is not transitive
• weak pronormality is not transitive
• Polynormality is not transitive
• Weak normality is not transitive

Facts used

1. Normal implies pronormal
2. Pronormal and subnormal implies normal
3. Normality is not transitive

Proof

By fact (3), construct subgroups $H\leq K\leq G$ such that $H$ is normal in $K$, $K$ is normal in $G$, but $H$ is not normal in $G$.

• By fact (1), $H$ is pronormal in $K$ and $K$ is pronormal in $G$.
• By definition, $H$ is subnormal in $G$, so by fact (2), if $H$ were pronormal in $G$, $H$ would also be normal in $G$. But by construction, $H$ is not normal in $G$, so $H$ is not pronormal in $G$.

In particular, any example showing that normality is not transitive also shows that pronormality is not transitive.