# Pronormality is not centralizer-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., centralizer-closed subgroup property).

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## Statement

It is possible to have a group and a pronormal subgroup of such that the centralizer is not pronormal.

## Related facts

- Pronormality is normalizer-closed: In fact, normalizer of pronormal implies abnormal
- Pronormality is not commutator-closed
- Automorph-conjugacy is centralizer-closed
- Isomorph-conjugacy is not centralizer-closed

## Proof

### Example of the symmetric group

`Further information: symmetric group:S4, dihedral group:D8`

Suppose is the symmetric group on the set and is a -Sylow subgroup of , given by:

.

Then, is a pronormal subgroup of , because it is a Sylow subgroup and Sylow implies pronormal. On the other hand, we have:

.

This is not pronormal, because it is conjugate to the subgroup via the permutation , but these are not conjugate in the subgroup they generate.

Let be the symmetric group on the set .