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Every nontrivial normal subgroup is potentially 2-subnormal-and-not-normal
From Groupprops
Contents |
Statement
Suppose G is a group and H is a nontrivial normal subgroup of G. Then, there exists a group K containing G such that H is a 2-subnormal subgroup of K but not a normal subgroup of K.
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
The relation with characteristic subgroups
- Characteristic of normal implies normal, left transiter of normal is characteristic
- Every nontrivial normal subgroup is potentially normal-and-not-characteristic
