# Pronormality is not commutator-closed

From Groupprops

This article gives the statement, and possibly proof, of a subgroup property (i.e., pronormal subgroup)notsatisfying a subgroup metaproperty (i.e., commutator-closed subgroup property).

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## Statement

It is possible to have a group and pronormal subgroups of such that the commutator is also a pronormal subgroup.

## Related facts

- Normality is commutator-closed
- Characteristicity is commutator-closed
- Endo-invariance implies commutator-closed
- Join of subnormal subgroups is subnormal iff their commutator is subnormal

## Proof

### Example of the symmetric group of degree four

`Further information: symmetric group:S4`

Let be the symmetric group on the set . Let be a -Sylow subgroup of , say:

.

Then, is isomorphic to a dihedral group of order eight. .

- is pronormal in : In fact, is a Sylow subgroup of , and Sylow implies pronormal.
- is not pronormal in : This subgroup is conjugate to by the permutation , but these two subgroups are not conjugate in the subgroup they generate.