,"Difficulty level","Fact about" "Cardinality of underlying set of a profinite group need not determine order as a profinite group",,"Order of a profinite group; 2,Order of a group; 3" "Conjecture that most finite groups are nilpotent",,"Finite group; ?,Order of a group; ?,Finite nilpotent group; ?" "Degree of irreducible representation divides order of group",4,"Degree of a linear representation; 1,Irreducible linear representation; 1,Order of a group; 1,Group; 1" "Exponent divides order in finite group",,"Exponent of a group; ?,Order of a group; ?" "Exponent of a finite group has precisely the same prime factors as order",2,"Order of a group; 1,Exponent of a group; 1" "Finitely many subgroups iff finite",,"Finite group; ?,Order of a group; ?" "Lagrange's theorem",2,"Order of a group; 1,Subgroup; 1,Group; 1,Size; 1,Left coset of a subgroup; 1" "Minimal normal subgroup with order greater than index is characteristic",,"Minimal normal subgroup; ?,Order of a group; ?,Index of a subgroup; ?,Characteristic subgroup; ?" "Minimal normal subgroup with order not dividing index is characteristic",,"Minimal normal subgroup; 2,Order of a group; 3,Index of a subgroup; 3,Characteristic subgroup; 3" "Order of a profinite group need not determine order as a group in the sense of cardinality of underlying set",,"Order of a profinite group; 2,Order of a group; 3" "Order of direct product is product of orders",0,"Order of a group; 2,External direct product; 2,Product; 2,Internal direct product; ?" "Order of element divides order of group",,"Order of an element; 1,Element of a group; 1,Order of a group; 1,Group; 1,Order of an element; ?" "Order of extension group is product of order of normal subgroup and quotient group",,"Order of a group; 2,Group extension; 1,Product; 2" "Order of quotient group divides order of group",,"Order of a group; 1,Quotient map; 1,Group; 1" "Order of semidirect product is product of orders",1,"Order of a group; 2,External semidirect product; 1,Product; 2,Internal semidirect product; 1" "Pyber's theorem on logarithmic quotient of number of nilpotent groups to number of groups approaching unity",,"Finite group; ?,Order of a group; ?,Finite nilpotent group; ?" "Size of conjugacy class divides order of inner automorphism group",,"Conjugacy class; ?,Order of a group; ?,Inner automorphism group; ?,Index of a subgroup; ?,Center; ?" "Square of Schur index of irreducible character in characteristic zero divides order",,"Irreducible linear representation; ?,Schur index of irreducible character; ?,Order of a group; ?"