# Pronormality does not satisfy transfer condition

From Groupprops

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

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## Contents

## Statement

It is possible to have a group , a pronormal subgroup of , and a subgroup of , such that is not pronormal in .

## Related facts

### Related metaproperties satisfied by pronormality

- Pronormality satisfies image condition
- Pronormality satisfies intermediate subgroup condition
- Pronormality is quotient-transitive

### Related metaproperties not satisfied by pronormality

### Other facts relating pronormality and the transfer condition

## Related properties

- Transfer-closed pronormal subgroup: A subgroup whose intersection with any other subgroup is pronormal in that other subgroup.

## Facts used

## Proof

### Example of the symmetric group

`Further information: symmetric group:S4`

Let be the symmetric group on the set . Consider subgroups of as follows:

.

Then, we have:

.

Note that:

- is not pronormal in : Indeed, the subgroup is conjugate to it in , but not in the subgroup they generate. Another way of seeing this is that is 2-subnormal but not normal in , hence it cannot be pronormal in .
- is pronormal in : is a subgroup whose join with any distinct conjugate is the whole group. Thus, by fact (1), is pronormal in .