# Semantic search

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, Central factor implies normal, Central 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, Comparable with all normal subgroups implies normal in nilpotent group, Composition of subgroup property satisfying intermediate subgroup condition with normality equals property in normal closure, Conjugate-comparable not implies normal, Conjunction of normality with any nontrivial finite-direct product-closed property of groups is not transitive, Direct factor implies normal, Equivalence of conjugacy and commutator definitions of normality, Equivalence of conjugacy and coset definitions of normality, 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, Extensible automorphism-invariant equals normal, Finitary symmetric group is normal in symmetric group, First isomorphism theorem, Fourth isomorphism theorem, Index three implies normal or double coset index two, Induced class function from normal subgroup is zero outside the subgroup, Inner automorphism to automorphism is right tight for normality, Intermediately automorph-conjugate of normal implies weakly pronormal, Intermediately isomorph-conjugate of normal implies pronormal, Intermediately normal-to-characteristic of normal implies intermediately subnormal-to-normal, Join of normal and subnormal implies subnormal of same depth, Join-transitively subnormal of normal implies finite-conjugate-join-closed subnormal, Left residual of 2-subnormal by normal is normal of characteristic, Left transiter of normal is characteristic, Left-transitively WNSCDIN not implies normal, Maximal conjugate-permutable implies normal, Maximal implies normal or abnormal, Maximal implies normal or self-normalizing, Maximal permutable implies normal, Nilpotent implies every maximal subgroup is normal, Nilpotent implies every normal subgroup is potentially characteristic, No subgroup property between normal Sylow and subnormal or between Sylow retract and retract is conditionally lattice-determined, Normal and self-centralizing implies coprime automorphism-faithful, Normal and self-centralizing implies normality-large, Normal equals potentially characteristic