# Finite solvable not implies p-normal

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., finite solvable group) neednotsatisfy the second subgroup property (i.e., p-normal group)

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## Statement

It is possible to have a finite solvable group and a prime such that is not a p-normal group. In other words, there exists a -Sylow subgroup of whose center is not weakly closed in it.

## Proof

### Example of the symmetric group of degree four

`Further information: symmetric group:S4`

Let be the symmetric group on the set and let . Then:

- is solvable.
- is not -normal: The center of a -Sylow subgroup is not weakly closed in it. For instance, consider the -Sylow subgroup:

.

The center of this is:

.

This is not weakly closed in . For instance, it is conjugate in to the subgroup generated by , which is another subgroup inside .