# Sylow not implies NE

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., Sylow subgroup) neednotsatisfy the second subgroup property (i.e., NE-subgroup)

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

Get more facts about Sylow subgroup|Get more facts about NE-subgroup

EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property Sylow subgroup but not NE-subgroup|View examples of subgroups satisfying property Sylow subgroup and NE-subgroup

## Contents

## Statement

A Sylow subgroup of a finite group need not be a NE-subgroup: it need not equal the intersection of its normalizer and normal closure in the whole group.

## Facts used

## Related facts

## Proof

### Example of the alternating group of degree five

`Further information: alternating group:A5`

Let be the alternating group on . Then:

- Let be a -Sylow subgroup of , for instance, . Since is simple, the normal closure of in is . The normalizer of in is the alternating group on , which is strictly bigger than . Thus, the intersection of the normalizer and normal closure is strictly bigger than .
- Let be a -Sylow subgroup of , for instance, . Then, the normalizer of in is a dihedral group of order ten, which in particular includes double transpositions such as . The normal closure of in is . Thus, the intersection of the normalizer and normal closure is strictly bigger than .