# Hall not implies pronormal

From Groupprops

This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., Hall subgroup) neednotsatisfy the second subgroup property (i.e., pronormal subgroup)

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## Statement

We can have a finite group and a Hall subgroup of such that is not a pronormal subgroup of .

## Facts used

## Related facts

## Proof

### General proof

By fact (1), construct a finite group and Hall subgroups of such that and are Hall of the same order but are not conjugate.

Let be a prime that does not divide the order of . Consider the wreath product of with the cyclic group of order acting regularly. This is a group given by:

.

Now consider the subgroups of given by:

.

- is a Hall subgroup of : is clearly a Hall subgroup of . is a Hall subgroup of because is a Hall subgroup of (fact (2)).
- is not pronormal in : Consider and its conjugate by a generator of the . We have . These are both subgroups in , hence if they are conjugate in the subgroup they are generate, they are conjugate in . However, if and are conjugate in , then the conjugating element acts coordinate-wise, so the first coordinate of (which is ) is conjugate to the first coordinate of (which is ) in . But this is contradictory to the assumption that and are not conjugate in .