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 $H \le K \le G$ 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

Other instances of the generalization include:

Proof

By fact (3), construct subgroups $H \le K \le G$ 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.