Automorph-join-closed subnormal of normal implies conjugate-join-closed subnormal | 1 | Composition operator (?) Automorph-join-closed subnormal subgroup (?) Normal subgroup (?) Conjugate-join-closed subnormal subgroup (?) |

Automorph-permutable of normal implies conjugate-permutable | 1 | 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 | 4 | Normal subgroup (?) Division ring (2) |

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

Central implies normal | 1 | Central subgroup (2) Normal subgroup (2) |

Characteristic implies normal | 1 | Characteristic subgroup (1) Normal subgroup (1) |

Characteristic of normal implies normal | 1 | Composition operator (?) Characteristic subgroup (?) Normal subgroup (?) |

Commutator of a group and a subgroup implies normal | 3 | 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 | 3 | 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 | | 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 | 1 | Normal subgroup (1) |

Equivalence of conjugacy and coset definitions of normality | 1 | 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 | 0 | 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 | | Normal subgroup (?) Characteristic subgroup (?) |

Every nontrivial normal subgroup is potentially normal-and-not-characteristic | | Normal subgroup (?) |

Every subgroup is contracharacteristic in its normal closure | | Composition operator (?) Contracharacteristic subgroup (?) Normal subgroup (?) Subgroup (?) |

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

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

First isomorphism theorem | 2 | Normal subgroup (?) |

Fourth isomorphism theorem | 2 | 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 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 | | 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 operator (?) Subgroup property (?) Normal subgroup (?) Subnormal subgroup (?) Subnormal depth (2) |

Join-transitively 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 (2) Normal subgroup (3) 2-subnormal subgroup (3) |

Left transiter of normal is characteristic | 3 | Characteristic subgroup (?) Normal subgroup (?) |

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

Maximal conjugate-permutable implies normal | | Normal subgroup (?) Conjugate-permutable subgroup (?) |

Maximal implies normal or abnormal | 2 | Maximal subgroup (?) Normal subgroup (?) Abnormal subgroup (?) |

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

Maximal permutable implies normal | | Normal subgroup (?) Permutable subgroup (?) |

Nilpotent 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 | | 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 | | 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 | 3 | Normal subgroup (1) |

Normal equals retract-potentially characteristic | | Normal subgroup (1) |

Normal equals strongly image-potentially characteristic | | Normal subgroup (1) |

Normal iff potential endomorphism kernel | | Normal subgroup (1) |

Normal implies join-transitively subnormal | | Normal subgroup (2) Join-transitively subnormal subgroup (2) |

Normal implies modular | | Normal subgroup (2) Modular subgroup (2) |

Normal implies permutable | 1 | Normal subgroup (2) Permutable subgroup (2) Normal subgroup (?) Permutable subgroup (?) Product of subgroups (?) |

Normal not implies amalgam-characteristic | 3 | Normal subgroup (2) Amalgam-characteristic subgroup (1) |

Normal not implies central factor | | Normal subgroup (2) Central factor (2) |

Normal not implies characteristic | 2 | Normal subgroup (1) Characteristic subgroup (1) |

Normal not implies characteristic in the collection of all groups satisfying a nontrivial finite direct product-closed group property | 3 | Normal subgroup (?) Characteristic subgroup (?) |

Normal not implies direct factor | 2 | Normal subgroup (2) Direct factor (2) |

Normal not implies finite-pi-potentially characteristic in finite | | Finite group (2) Normal subgroup (2) Finite-pi-potentially characteristic subgroup (2) |

Normal not implies image-potentially fully invariant | | Normal subgroup (2) Image-potentially fully invariant subgroup (2) |

Normal not implies left-transitively fixed-depth subnormal | 3 | Normal subgroup (2) Left-transitively fixed-depth subnormal subgroup (2) Subnormal subgroup (2) |

Normal not implies normal-extensible automorphism-invariant in finite | | Finite group (2) Normal subgroup (2) Normal-extensible automorphism-invariant subgroup (2) |

Normal not implies normal-potentially characteristic | | Normal subgroup (2) Normal-potentially characteristic subgroup (2) |

Normal not implies normal-potentially relatively characteristic | | Normal subgroup (2) Normal-potentially relatively characteristic subgroup (2) |

Normal not implies potentially fully invariant | | Normal subgroup (2) Potentially fully invariant subgroup (2) |

Normal not implies potentially verbal | | Normal subgroup (2) Potentially verbal subgroup (2) |

Normal not implies right-transitively fixed-depth subnormal | | Normal subgroup (2) Right-transitively fixed-depth subnormal subgroup (2) Subnormal subgroup (2) |

Normal not implies strongly potentially characteristic | | Normal subgroup (2) Characteristic-potentially characteristic subgroup (2) |

Normal of order equal to least prime divisor of group order implies central | | Normal subgroup (?) |

Normal of order two implies central | | Normal subgroup (?) Cyclic group:Z2 (?) Central subgroup (?) |

Normal subgroup equals kernel of homomorphism | | Normal subgroup (1) Homomorphism of groups (?) Homomorphism of groups (?) Normal subgroup (?) |

Normal subgroup of ambivalent group implies every element is automorphic to its inverse | | Ambivalent group (?) Normal subgroup (?) Group in which every element is automorphic to its inverse (?) |

Normal subset generates normal subgroup | | Normal subset of a group (1) Normal subgroup (3) |

Normal upper-hook fully normalized implies characteristic | | Normal subgroup (?) Fully normalized subgroup (?) Characteristic subgroup (?) |

Normal-extensible not implies normal | | Normal-extensible automorphism (2) Normal automorphism (2) Normal subgroup (2) Normal-extensible automorphism-invariant subgroup (2) |

Normality is commutator-closed | 1 | Normal subgroup (1) Commutator-closed subgroup property (2) |

Normality is not transitive | 2 | Normal subgroup (1) Transitive subgroup property (2) Base of a wreath product (?) |

Normality is not transitive for any pair of nontrivial quotient groups | | Normal subgroup (?) |

Normality is preserved under any monotone subgroup-defining function | | Normal subgroup (3) |

Normality is strongly UL-intersection-closed | 1 | Normal subgroup (1) Strongly UL-intersection-closed subgroup property (2) |

Normality is strongly intersection-closed | 1 | Normal subgroup (1) Strongly intersection-closed subgroup property (2) Intersection of subgroups (?) Normal subgroup (?) |

Normality is strongly join-closed | 1 | Normal subgroup (?) Join of subgroups (?) |

Normality is upper join-closed | 1 | Normal subgroup (1) Upper join-closed subgroup property (2) |

Normality satisfies image condition | 1 | Normal subgroup (1) Image condition (2) |

Normality satisfies intermediate subgroup condition | 1 | Normal subgroup (1) Intermediate subgroup condition (2) Normal subgroup (?) Intermediate subgroup condition (?) |

Normality satisfies lower central series condition | 1 | Normal subgroup (1) Lower central series condition (2) |

Normality satisfies partition difference condition | 1 | Normal subgroup (1) Partition difference condition (2) |

Permutable not implies normal | 3 | Permutable subgroup (2) Normal subgroup (2) |

Potentially normal-subhomomorph-containing equals normal | | Normal subgroup (1) Normal-subhomomorph-containing subgroup (?) Normal-homomorph-containing subgroup (?) Strictly characteristic subgroup (?) |

Potentially verbal implies normal | | Potentially verbal subgroup (2) Normal subgroup (2) |

Procharacteristic of normal implies pronormal | | Composition operator (?) Procharacteristic subgroup (?) Normal subgroup (?) Pronormal subgroup (?) |

Proper and normal in quasisimple implies central | | Normal subgroup (?) Quasisimple group (2) Group in which every proper normal subgroup is central (2) |

Quotient group need not be isomorphic to any subgroup | | Normal subgroup (2) Endomorphism kernel (2) Subgroup (2) Quotient map (2) |

Second isomorphism theorem | 2 | Normal subgroup (2) |

Subgroup isomorphic to whole group need not be normal | | Normal subgroup (?) |

Subgroup of index equal to least prime divisor of group order is normal | | Subgroup of prime index (?) Index of a subgroup (?) Normal subgroup (?) Finite group (?) Subgroup of index equal to least prime divisor of group order (?) Subgroup of index equal to least prime divisor of group order of finite group (2) Normal subgroup of finite group (3) |

Subgroup of index two is normal | | Subgroup of index two (2) Normal subgroup (2) |