# P-normal not implies p-solvable

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

View a complete list of subgroup property non-implications | View a complete list of subgroup property implications

Get more facts about p-normal group|Get more facts about p-solvable group

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## Statement

We can have a finite group and a prime number such that is a p-normal group but is not a p-solvable group.

## Facts used

## Proof

### Example of the alternating group of degree five

`Further information: alternating group:A5`

The alternating group of degree five is a -normal group for all primes , because all its Sylow subgroups are abelian. On the other hand, since the group is simple non-abelian, it is not -solvable for .