[[Fact about::Normal subgroup]]

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	Uses	Fact about
Automorph-join-closed subnormal of normal implies conjugate-join-closed subnormal		Composition operator Automorph-join-closed subnormal subgroup Normal subgroup Conjugate-join-closed subnormal subgroup
Automorph-permutable of normal implies conjugate-permutable		Composition operator Automorph-permutable subgroup Normal subgroup Conjugate-permutable subgroup
Bound on double coset index in terms of orders of group and subgroup		Double coset of a pair of subgroups Double coset index of a subgroup Subgroup of finite index Malnormal subgroup Normal subgroup Frobenius subgroup
Cartan-Brauer-Hua theorem		Normal subgroup
Center is normal		Center Normal subgroup
Central factor implies normal		Central factor Normal subgroup
Characteristic implies normal	Group acts as automorphisms by conjugation	Characteristic subgroup Normal subgroup
Characteristic of normal implies normal		Composition operator Characteristic subgroup Normal subgroup
Commutator of a group and a subgroup implies normal	Subgroup normalizes its commutator with any subset	Composition operator Subgroup property Improper subgroup Subgroup Normal subgroup Subgroup realizable as the commutator of the whole group and a subgroup
Commutator of a group and a subset implies normal	Subgroup normalizes its commutator with any subset	Composition operator Subgroup property Improper subgroup Subset of a group Normal subgroup
Commutator of a normal subgroup and a subgroup not implies normal		Normal subgroup Commutator of two subgroups
Commutator of a normal subgroup and a subset implies 2-subnormal	Subgroup normalizes its commutator with any subset	Composition operator Subgroup property Normal subgroup Subset of a group 2-subnormal subgroup Subgroup realizable as the commutator of a normal subgroup and a subset
Composition of subgroup property satisfying intermediate subgroup condition with normality equals property in normal closure		Intermediate subgroup condition Composition operator Normal subgroup Normal closure
Conjunction of normality with any nontrivial finite-direct product-closed property of groups is not transitive		Finite-direct product-closed group property Normal subgroup
Every group is normal fully normalized in its holomorph		Holomorph of a group Normal subgroup Fully normalized subgroup
Every group is normal in itself		Normal subgroup Identity-true subgroup property
Every nontrivial normal subgroup is potentially characteristic-and-not-fully invariant		Normal subgroup Characteristic subgroup Fully invariant subgroup
Every nontrivial normal subgroup is potentially characteristic-and-not-intermediately characteristic	Every nontrivial normal subgroup is potentially normal-and-not-characteristic NPC theorem	Normal subgroup Characteristic subgroup
Every nontrivial normal subgroup is potentially normal-and-not-characteristic		Normal subgroup
Every subgroup is contracharacteristic in its normal closure	Characteristic of normal implies normal	Composition operator Contracharacteristic subgroup Normal subgroup Subgroup

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