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 (?) Division ring (2) |

Central factor implies normal | | Central factor (2) Normal subgroup (2) |

Central implies normal | | Central subgroup (2) Normal subgroup (2) |

Characteristic implies normal | Group acts as automorphisms by conjugation | Characteristic subgroup (1) Normal subgroup (1) |

Characteristic of normal implies normal | Restriction of automorphism to subgroup invariant under it and its inverse is automorphism Composition rule for function restriction | Composition operator (?) Characteristic subgroup (?) Normal subgroup (?) |

Commutator of a group and a subgroup implies normal | Subgroup normalizes its commutator with any subset | Commutator operator (3) Improper subgroup (2) Subgroup (2) Normal subgroup (2) Subgroup realizable as the commutator of the whole group and a subgroup (2) |

Commutator of a group and a subset implies normal | Subgroup normalizes its commutator with any subset | Commutator operator (3) Improper subgroup (2) Subset of a group (2) Normal subgroup (2) |

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 | Commutator operator (3) Normal subgroup (2) Subset of a group (2) 2-subnormal subgroup (2) Subgroup realizable as the commutator of a normal subgroup and a subset (2) |

Comparable with all normal subgroups implies normal in nilpotent group | | Nilpotent group (?) Subgroup comparable with all normal subgroups (?) Normal subgroup (?) Subgroup comparable with all normal subgroups of nilpotent group (2) Normal subgroup of nilpotent group (3) |

Composition of subgroup property satisfying intermediate subgroup condition with normality equals property in normal closure | | Intermediate subgroup condition (?) Composition operator (?) Normal subgroup (?) Normal closure (?) |

Conjugate-comparable not implies normal | | Conjugate-comparable subgroup (2) Normal subgroup (2) |

Conjunction of normality with any nontrivial finite-direct product-closed property of groups is not transitive | | Finite-direct product-closed group property (?) Normal subgroup (?) |

Direct factor implies normal | | Direct factor (2) Normal subgroup (2) |

Equivalence of conjugacy and commutator definitions of normality | | Normal subgroup (1) |

Equivalence of conjugacy and coset definitions of normality | | Normal subgroup (1) Normalizer of a subgroup (1) Conjugate subgroups (?) Left coset of a subgroup (?) Normalizer of a subgroup (?) 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 (1) Identity-true subgroup property (2) |

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 Normal equals potentially characteristic | 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 (?) |

Extensible automorphism-invariant equals normal | Inner implies extensible Extensible implies normal | Normal subgroup (1) |

Finitary symmetric group is normal in symmetric group | | Finitary symmetric group (?) Normal subgroup (?) Symmetric group (?) |

First isomorphism theorem | Normal subgroup equals kernel of homomorphism | Normal subgroup (?) |

Fourth isomorphism theorem | | Normal subgroup (?) |

Index three implies normal or double coset index two | | Subgroup of index three (?) Normal subgroup (?) Subgroup of double coset index two (?) |

Induced class function from normal subgroup is zero outside the subgroup | | Normal subgroup (?) Class function (?) Induced class function (?) |

Inner automorphism to automorphism is right tight for normality | | Normal subgroup (?) |

Intermediately automorph-conjugate of normal implies weakly pronormal | | Intermediately automorph-conjugate subgroup of normal subgroup (2) Weakly pronormal subgroup (2) Intermediately automorph-conjugate subgroup (?) Normal subgroup (?) |

Intermediately isomorph-conjugate of normal implies pronormal | Intermediately isomorph-conjugate implies procharacteristic Procharacteristic of normal implies pronormal | Intermediately isomorph-conjugate subgroup of normal subgroup (2) Pronormal subgroup (2) Composition operator (?) Intermediately isomorph-conjugate subgroup (?) Normal subgroup (?) Pronormal subgroup (?) |

Intermediately normal-to-characteristic of normal implies intermediately subnormal-to-normal | Subnormality of fixed depth satisfies intermediate subgroup condition Intermediately normal-to-characteristic implies intermediately subnormal-to-normal Normality satisfies transfer condition Characteristic of normal implies normal | Composition operator (?) Intermediately normal-to-characteristic subgroup (?) Normal subgroup (?) Intermediately subnormal-to-normal subgroup (?) |

Join of normal and subnormal implies subnormal of same depth | Join lemma for normal subgroup of subgroup with normal subgroup of whole group | Join operator (?) Subgroup property (?) Normal subgroup (?) Subnormal subgroup (?) Subnormal depth (2) |

Join-transitively subnormal of normal implies finite-conjugate-join-closed subnormal | Join-transitively subnormal implies finite-automorph-join-closed subnormal Finite-automorph-join-closed subnormal of normal implies finite-conjugate-join-closed subnormal | Composition operator (?) Join-transitively subnormal subgroup (?) Normal subgroup (?) Finite-conjugate-join-closed subnormal subgroup (?) |

Left residual of 2-subnormal by normal is normal of characteristic | | Normal subgroup of characteristic subgroup (1) Normal subgroup of characteristic subgroup (?) Normal subgroup (?) 2-subnormal subgroup (?) |

Left transiter of normal is characteristic | Characteristic of normal implies normal Inner automorphism to automorphism is right tight for normality | Characteristic subgroup (?) Normal subgroup (?) |

Left-transitively WNSCDIN not implies normal | | Left-transitively WNSCDIN-subgroup (2) Normal subgroup (2) |

Maximal conjugate-permutable implies normal | Conjugate-permutability is conjugate-join-closed Product of conjugates is proper | Normal subgroup (?) Conjugate-permutable subgroup (?) |

Maximal implies normal or abnormal | | Maximal subgroup (?) Normal subgroup (?) Abnormal subgroup (?) |

Maximal implies normal or self-normalizing | | Maximal subgroup (?) Normal subgroup (?) Self-normalizing subgroup (?) |

Maximal permutable implies normal | Permutability is strongly join-closed Product of conjugates is proper | Normal subgroup (?) Permutable subgroup (?) |

Nilpotent implies every maximal subgroup is normal | Nilpotent implies normalizer condition Normalizer condition implies every maximal subgroup is normal | Nilpotent group (?) Maximal subgroup (?) Normal subgroup (?) Maximal subgroup of nilpotent group (2) Normal subgroup of nilpotent group (3) Group in which every maximal subgroup is normal (?) |

Nilpotent implies every normal subgroup is potentially characteristic | Normal subgroup contained in hypercenter is potentially characteristic | Nilpotent group (?) Normal subgroup (?) Potentially characteristic subgroup (?) Normal subgroup of nilpotent group (2) Potentially characteristic subgroup of nilpotent group (3) |

No subgroup property between normal Sylow and subnormal or between Sylow retract and retract is conditionally lattice-determined | | Subnormal subgroup (1) Conditionally lattice-determined subgroup property (2) Normal subgroup (1) Characteristic subgroup (1) Fully invariant subgroup (1) Normal Hall subgroup (1) Normal Sylow subgroup (1) Complemented normal subgroup (1) Sylow retract (1) Hall retract (1) Retract (1) |

Normal and self-centralizing implies coprime automorphism-faithful | Three subgroup lemma Stability group of subnormal series of finite group has no other prime factors | Self-centralizing normal subgroup (2) Coprime automorphism-faithful normal subgroup (2) Coprime automorphism group (?) Normal subgroup (?) Self-centralizing subgroup (?) Coprime automorphism-faithful subgroup (?) |

Normal and self-centralizing implies normality-large | | Self-centralizing normal subgroup (2) Normality-large normal subgroup (2) Normal subgroup (?) Self-centralizing subgroup (?) Normality-large subgroup (?) |

Normal equals potentially characteristic | Characteristicity is centralizer-closed Characteristic implies normal Normality satisfies intermediate subgroup condition | Normal subgroup (1) |