# Every nontrivial normal subgroup is potentially 2-subnormal-and-not-normal

From Groupprops

## Contents

## Statement

Suppose is a group and is a nontrivial normal subgroup of . Then, there exists a group containing such that is a 2-subnormal subgroup of but not a normal subgroup of .

## Related facts

### About the lack of transitivity of normality

- Normality is not transitive
- Normality is not transitive in any nontrivial extension-closed subquasivariety of the quasivariety of groups
- Conjunction of normality with any nontrivial finite-direct product-closed property of groups is not transitive
- There exist subgroups of arbitrarily large subnormal depth