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The query [[Fact about.Page::Normal subgroup]] was answered by the SMWSQLStore3 in 0.0125 seconds.


Results 1 – 100    (Previous 100 | Next 100)   (20 | 50 | 100 | 250 | 500)   (JSON | CSV | RSS | RDF)
 Difficulty levelFact about
Automorph-join-closed subnormal of normal implies conjugate-join-closed subnormal1Composition operator (?)
Automorph-join-closed subnormal subgroup (?)
Normal subgroup (?)
Conjugate-join-closed subnormal subgroup (?)
Automorph-permutable of normal implies conjugate-permutable1Composition operator (?)
Automorph-permutable subgroup (?)
Normal subgroup (?)
Conjugate-permutable subgroup (?)
Bound on double coset index in terms of orders of group and subgroupDouble 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 theorem4Normal subgroup (?)
Division ring (2)
Central factor implies normalCentral factor (2)
Normal subgroup (2)
Central implies normal1Central subgroup (2)
Normal subgroup (2)
Characteristic implies normal1Characteristic subgroup (1)
Normal subgroup (1)
Characteristic of normal implies normal1Composition operator (?)
Characteristic subgroup (?)
Normal subgroup (?)
Commutator of a group and a subgroup implies normal3Commutator 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 normal3Commutator operator (3)
Improper subgroup (2)
Subset of a group (2)
Normal subgroup (2)
Commutator of a normal subgroup and a subgroup not implies normalNormal subgroup (?)
Commutator of two subgroups (?)
Commutator of a normal subgroup and a subset implies 2-subnormalCommutator 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 groupNilpotent 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 closureIntermediate subgroup condition (?)
Composition operator (?)
Normal subgroup (?)
Normal closure (?)
Conjugate-comparable not implies normalConjugate-comparable subgroup (2)
Normal subgroup (2)
Conjunction of normality with any nontrivial finite-direct product-closed property of groups is not transitiveFinite-direct product-closed group property (?)
Normal subgroup (?)
Direct factor implies normalDirect factor (2)
Normal subgroup (2)
Equivalence of conjugacy and commutator definitions of normality1Normal subgroup (1)
Equivalence of conjugacy and coset definitions of normality1Normal 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 holomorphHolomorph of a group (?)
Normal subgroup (?)
Fully normalized subgroup (?)
Every group is normal in itself0Normal subgroup (1)
Identity-true subgroup property (2)
Every nontrivial normal subgroup is potentially characteristic-and-not-fully invariantNormal subgroup (?)
Characteristic subgroup (?)
Fully invariant subgroup (?)
Every nontrivial normal subgroup is potentially characteristic-and-not-intermediately characteristicNormal subgroup (?)
Characteristic subgroup (?)
Every nontrivial normal subgroup is potentially normal-and-not-characteristicNormal subgroup (?)
Every subgroup is contracharacteristic in its normal closureComposition operator (?)
Contracharacteristic subgroup (?)
Normal subgroup (?)
Subgroup (?)
Extensible automorphism-invariant equals normalNormal subgroup (1)
Finitary symmetric group is normal in symmetric groupFinitary symmetric group (?)
Normal subgroup (?)
Symmetric group (?)
First isomorphism theorem2Normal subgroup (?)
Fourth isomorphism theorem2Normal subgroup (?)
Index three implies normal or double coset index twoSubgroup of index three (?)
Normal subgroup (?)
Subgroup of double coset index two (?)
Induced class function from normal subgroup is zero outside the subgroupNormal subgroup (?)
Class function (?)
Induced class function (?)
Inner automorphism to automorphism is right tight for normalityNormal subgroup (?)
Intermediately automorph-conjugate of normal implies weakly pronormalIntermediately automorph-conjugate subgroup of normal subgroup (2)
Weakly pronormal subgroup (2)
Intermediately automorph-conjugate subgroup (?)
Normal subgroup (?)
Intermediately isomorph-conjugate of normal implies pronormalIntermediately 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-normalComposition operator (?)
Intermediately normal-to-characteristic subgroup (?)
Normal subgroup (?)
Intermediately subnormal-to-normal subgroup (?)
Join of normal and subnormal implies subnormal of same depthJoin operator (?)
Subgroup property (?)
Normal subgroup (?)
Subnormal subgroup (?)
Subnormal depth (2)
Join-transitively subnormal of normal implies finite-conjugate-join-closed subnormalComposition operator (?)
Join-transitively subnormal subgroup (?)
Normal subgroup (?)
Finite-conjugate-join-closed subnormal subgroup (?)
Left residual of 2-subnormal by normal is normal of characteristicNormal subgroup of characteristic subgroup (1)
Normal subgroup of characteristic subgroup (2)
Normal subgroup (3)
2-subnormal subgroup (3)
Left transiter of normal is characteristic3Characteristic subgroup (?)
Normal subgroup (?)
Left-transitively WNSCDIN not implies normalLeft-transitively WNSCDIN-subgroup (2)
Normal subgroup (2)
Maximal conjugate-permutable implies normalNormal subgroup (?)
Conjugate-permutable subgroup (?)
Maximal implies normal or abnormal2Maximal subgroup (?)
Normal subgroup (?)
Abnormal subgroup (?)
Maximal implies normal or self-normalizingMaximal subgroup (?)
Normal subgroup (?)
Self-normalizing subgroup (?)
Maximal permutable implies normalNormal subgroup (?)
Permutable subgroup (?)
Nilpotent implies every maximal subgroup is normalNilpotent 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 characteristicNilpotent 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-determinedSubnormal 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-faithfulSelf-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-largeSelf-centralizing normal subgroup (2)
Normality-large normal subgroup (2)
Normal subgroup (?)
Self-centralizing subgroup (?)
Normality-large subgroup (?)
Normal equals potentially characteristic3Normal subgroup (1)
Normal equals retract-potentially characteristicNormal subgroup (1)
Normal equals strongly image-potentially characteristicNormal subgroup (1)
Normal iff potential endomorphism kernelNormal subgroup (1)
Normal implies join-transitively subnormalNormal subgroup (2)
Join-transitively subnormal subgroup (2)
Normal implies modularNormal subgroup (2)
Modular subgroup (2)
Normal implies permutable1Normal subgroup (2)
Permutable subgroup (2)
Normal subgroup (?)
Permutable subgroup (?)
Product of subgroups (?)
Normal not implies amalgam-characteristic3Normal subgroup (2)
Amalgam-characteristic subgroup (1)
Normal not implies central factorNormal subgroup (2)
Central factor (2)
Normal not implies characteristic2Normal subgroup (1)
Characteristic subgroup (1)
Normal not implies characteristic in the collection of all groups satisfying a nontrivial finite direct product-closed group property3Normal subgroup (?)
Characteristic subgroup (?)
Normal not implies direct factor2Normal subgroup (2)
Direct factor (2)
Normal not implies finite-pi-potentially characteristic in finiteFinite group (2)
Normal subgroup (2)
Finite-pi-potentially characteristic subgroup (2)
Normal not implies image-potentially fully invariantNormal subgroup (2)
Image-potentially fully invariant subgroup (2)
Normal not implies left-transitively fixed-depth subnormal3Normal subgroup (2)
Left-transitively fixed-depth subnormal subgroup (2)
Subnormal subgroup (2)
Normal not implies normal-extensible automorphism-invariant in finiteFinite group (2)
Normal subgroup (2)
Normal-extensible automorphism-invariant subgroup (2)
Normal not implies normal-potentially characteristicNormal subgroup (2)
Normal-potentially characteristic subgroup (2)
Normal not implies normal-potentially relatively characteristicNormal subgroup (2)
Normal-potentially relatively characteristic subgroup (2)
Normal not implies potentially fully invariantNormal subgroup (2)
Potentially fully invariant subgroup (2)
Normal not implies potentially verbalNormal subgroup (2)
Potentially verbal subgroup (2)
Normal not implies right-transitively fixed-depth subnormalNormal subgroup (2)
Right-transitively fixed-depth subnormal subgroup (2)
Subnormal subgroup (2)
Normal not implies strongly potentially characteristicNormal subgroup (2)
Characteristic-potentially characteristic subgroup (2)
Normal of order equal to least prime divisor of group order implies centralNormal subgroup (?)
Normal of order two implies centralNormal subgroup (?)
Cyclic group:Z2 (?)
Central subgroup (?)
Normal subgroup equals kernel of homomorphismNormal subgroup (1)
Homomorphism of groups (?)
Homomorphism of groups (?)
Normal subgroup (?)
Normal subgroup of ambivalent group implies every element is automorphic to its inverseAmbivalent group (?)
Normal subgroup (?)
Group in which every element is automorphic to its inverse (?)
Normal subset generates normal subgroupNormal subset of a group (1)
Normal subgroup (3)
Normal upper-hook fully normalized implies characteristicNormal subgroup (?)
Fully normalized subgroup (?)
Characteristic subgroup (?)
Normal-extensible not implies normalNormal-extensible automorphism (2)
Normal automorphism (2)
Normal subgroup (2)
Normal-extensible automorphism-invariant subgroup (2)
Normality is commutator-closed1Normal subgroup (1)
Commutator-closed subgroup property (2)
Normality is not transitive2Normal subgroup (1)
Transitive subgroup property (2)
Base of a wreath product (?)
Normality is not transitive for any pair of nontrivial quotient groupsNormal subgroup (?)
Normality is preserved under any monotone subgroup-defining functionNormal subgroup (3)
Normality is strongly UL-intersection-closed1Normal subgroup (1)
Strongly UL-intersection-closed subgroup property (2)
Normality is strongly intersection-closed1Normal subgroup (1)
Strongly intersection-closed subgroup property (2)
Intersection of subgroups (?)
Normal subgroup (?)
Normality is strongly join-closed1Normal subgroup (?)
Join of subgroups (?)
Normality is upper join-closed1Normal subgroup (1)
Upper join-closed subgroup property (2)
Normality satisfies image condition1Normal subgroup (1)
Image condition (2)
Normality satisfies intermediate subgroup condition1Normal subgroup (1)
Intermediate subgroup condition (2)
Normal subgroup (?)
Intermediate subgroup condition (?)
Normality satisfies lower central series condition1Normal subgroup (1)
Lower central series condition (2)
Normality satisfies partition difference condition1Normal subgroup (1)
Partition difference condition (2)
Permutable not implies normal3Permutable subgroup (2)
Normal subgroup (2)
Potentially normal-subhomomorph-containing equals normalNormal subgroup (1)
Normal-subhomomorph-containing subgroup (?)
Normal-homomorph-containing subgroup (?)
Strictly characteristic subgroup (?)
Potentially verbal implies normalPotentially verbal subgroup (2)
Normal subgroup (2)
Procharacteristic of normal implies pronormalComposition operator (?)
Procharacteristic subgroup (?)
Normal subgroup (?)
Pronormal subgroup (?)
Proper and normal in quasisimple implies centralNormal subgroup (?)
Quasisimple group (2)
Group in which every proper normal subgroup is central (2)
Quotient group need not be isomorphic to any subgroupNormal subgroup (2)
Endomorphism kernel (2)
Subgroup (2)
Quotient map (2)
Second isomorphism theorem2Normal subgroup (2)
Subgroup isomorphic to whole group need not be normalNormal subgroup (?)
Subgroup of index equal to least prime divisor of group order is normalSubgroup 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 normalSubgroup of index two (2)
Normal subgroup (2)