# Semantic search

2-Sylow subgroup is TI implies it is normal or there is exactly one conjugacy class of involutions, 2-Sylow subgroup of rational group need not be rational, Automorphism group of finite nilpotent group is direct product of automorphism groups of Sylow subgroups, Burnside's normal p-complement theorem, Characteristic subgroup of Sylow subgroup is weakly closed iff it is normal in every Sylow subgroup containing it, Congruence condition on Sylow numbers, Congruence condition on number of Sylow subgroups containing a given subgroup of prime power order, Converse of congruence condition on Sylow numbers for the prime two, Divisibility condition on Sylow numbers, Every Sylow subgroup is cyclic implies metacyclic, Focal subgroup of a Sylow subgroup is generated by the commutators with normalizers of non-identity tame intersections, Focal subgroup theorem, Frattini's argument, Normal rank two Sylow subgroup for least prime divisor has normal complement if the prime is odd, Odd-order and normal rank two for all primes implies Sylow tower, Outer automorphism group of finite nilpotent group is direct product of outer automorphism groups of Sylow subgroups, Sylow does not satisfy transfer condition, Sylow implies MWNSCDIN, Sylow implies WNSCDIN, Sylow implies intermediately isomorph-conjugate, Sylow implies isomorph-conjugate, Sylow implies order-conjugate, Sylow implies order-dominated, Sylow implies order-dominating, Sylow implies pronormal, Sylow normalizer implies abnormal, Sylow not implies CDIN, Sylow not implies NE, Sylow not implies local divisibility-closed, Sylow number equals index of Sylow normalizer, Sylow of normal implies pronormal, Sylow satisfies intermediate subgroup condition, Sylow satisfies permuting transfer condition, Sylow subgroups are in correspondence with Sylow subgroups of quotient by central subgroup, Sylow subgroups are in correspondence with Sylow subgroups of quotient by hypercenter, Sylow subgroups exist, Symmetric group on finite set has Sylow subgroup of prime order