,"Difficulty level","Fact about"
"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"