,"Difficulty level","Fact about"
"Characteristic implies normal",1,"Characteristic subgroup; 1,Normal subgroup; 1"
"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 in itself",0,"Normal subgroup; 1,Identity-true subgroup property; 2"
"Extensible automorphism-invariant equals normal",,"Normal subgroup; 1"
"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 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 not implies characteristic",2,"Normal subgroup; 1,Characteristic subgroup; 1"
"Normal subgroup equals kernel of homomorphism",,"Normal subgroup; 1,Homomorphism of groups; ?,Homomorphism of groups; ?,Normal subgroup; ?"
"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 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 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"
"Potentially normal-subhomomorph-containing equals normal",,"Normal subgroup; 1,Normal-subhomomorph-containing subgroup; ?,Normal-homomorph-containing subgroup; ?,Strictly characteristic subgroup; ?"
"Trivial subgroup is normal",0,"Normal subgroup; 1,Trivially true subgroup property; 2"