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 Difficulty levelFact about
Cardinality of underlying set of a profinite group need not determine order as a profinite groupOrder of a profinite group (2)
Order of a group (3)
Conjecture that most finite groups are nilpotentFinite group (?)
Order of a group (?)
Finite nilpotent group (?)
Degree of irreducible representation divides order of group4Degree of a linear representation (1)
Irreducible linear representation (1)
Order of a group (1)
Group (1)
Exponent divides order in finite groupExponent of a group (?)
Order of a group (?)
Exponent of a finite group has precisely the same prime factors as order2Order of a group (1)
Exponent of a group (1)
Finitely many subgroups iff finiteFinite group (?)
Order of a group (?)
Lagrange's theorem2Order 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 characteristicMinimal normal subgroup (?)
Order of a group (?)
Index of a subgroup (?)
Characteristic subgroup (?)
Minimal normal subgroup with order not dividing index is characteristicMinimal 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 setOrder of a profinite group (2)
Order of a group (3)
Order of direct product is product of orders0Order of a group (2)
External direct product (2)
Product (2)
Internal direct product (?)
Order of element divides order of groupOrder 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 groupOrder of a group (2)
Group extension (1)
Product (2)
Order of quotient group divides order of groupOrder of a group (1)
Quotient map (1)
Group (1)
Order of semidirect product is product of orders1Order 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 unityFinite group (?)
Order of a group (?)
Finite nilpotent group (?)
Size of conjugacy class divides order of inner automorphism groupConjugacy class (?)
Order of a group (?)
Inner automorphism group (?)
Index of a subgroup (?)
Center (?)
Square of Schur index of irreducible character in characteristic zero divides orderIrreducible linear representation (?)
Schur index of irreducible character (?)
Order of a group (?)
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