# 2-subnormal not implies hypernormalized

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., 2-subnormal subgroup) neednotsatisfy the second subgroup property (i.e., hypernormalized subgroup)

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

Get more facts about 2-subnormal subgroup|Get more facts about hypernormalized subgroup

EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property 2-subnormal subgroup but not hypernormalized subgroup|View examples of subgroups satisfying property 2-subnormal subgroup and hypernormalized subgroup

## Contents

## Statement

### Verbal statement

A 2-subnormal subgroup of a group need not be hypernormalized.

## Related facts

### Stronger facts

- Abnormal normalizer and 2-subnormal not implies normal: In fact, the same example used here works for that as well.

## Proof

### An example in the symmetric group on four letters

Let be the symmetric group on four letters and be the two-element subgroup generated by .

Then, is normal in the subgroup , which is normal in . So is 2-subnormal in .

On the other hand, the normalizer is a dihedral subgroup of order eight, which is a self-normalizing subgroup.