]> 2022-05-20T03:49:59+00:00 Cardinality of underlying set of a profinite group need not determine order as a profinite group 0 en Conjecture that most finite groups are nilpotent 0 en Degree of irreducible representation divides order of group 0 en 4 Exponent divides order in finite group 0 en Exponent of a finite group has precisely the same prime factors as order 0 en 2 Finitely many subgroups iff finite 0 en Lagrange's theorem 0 en 2 Minimal normal subgroup with order greater than index is characteristic 0 en Minimal normal subgroup with order not dividing index is characteristic 0 en Order of a profinite group need not determine order as a group in the sense of cardinality of underlying set 0 en Order of direct product is product of orders 0 en 0 Order of element divides order of group 0 en Order of extension group is product of order of normal subgroup and quotient group 0 en Order of quotient group divides order of group 0 en Order of semidirect product is product of orders 0 en 1 Pyber's theorem on logarithmic quotient of number of nilpotent groups to number of groups approaching unity 0 en Size of conjugacy class divides order of inner automorphism group 0 en Square of Schur index of irreducible character in characteristic zero divides order 0 en