[[Fact about::Normal subgroup]]

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Automorph-join-closed subnormal of normal implies conjugate-join-closed subnormal
Automorph-permutable of normal implies conjugate-permutable
Bound on double coset index in terms of orders of group and subgroup
Cartan-Brauer-Hua theorem
Center is normal
Central factor implies normal
Characteristic implies normal
Characteristic of normal implies normal
Commutator of a group and a subgroup implies normal
Commutator of a group and a subset implies normal
Commutator of a normal subgroup and a subgroup not implies normal
Commutator of a normal subgroup and a subset implies 2-subnormal
Composition of subgroup property satisfying intermediate subgroup condition with normality equals property in normal closure
Conjunction of normality with any nontrivial finite-direct product-closed property of groups is not transitive
Every group is normal fully normalized in its holomorph
Every group is normal in itself
Every nontrivial normal subgroup is potentially characteristic-and-not-fully invariant
Every nontrivial normal subgroup is potentially characteristic-and-not-intermediately characteristic
Every nontrivial normal subgroup is potentially normal-and-not-characteristic
Every subgroup is contracharacteristic in its normal closure

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