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