]> 2019-07-21T14:43:20+00:00 2-Sylow subgroup of rational group is rational if its class is at most two 0 en Abelian direct factor implies potentially verbal in finite 0 en Central implies finite-pi-potentially verbal in finite 0 en Central implies potentially fully invariant in finite 0 en Central implies potentially verbal in finite 0 en Characteristic not implies isomorph-free in finite group 0 en Characteristic not implies isomorph-normal in finite group 0 en Characteristic not implies normal-isomorph-free 0 en Characteristic not implies sub-(isomorph-normal characteristic) in finite 0 en Characteristic not implies sub-isomorph-free in finite group 0 en Conjecture that most finite groups are nilpotent 0 en Conjugate-permutable implies subnormal in finite 0 en Cyclic normal implies finite-pi-potentially verbal in finite 0 en Cyclic normal implies potentially verbal in finite 0 en Finite group implies cyclic iff every subgroup is characteristic 0 en Finite groups are 1-isomorphic iff their directed power graphs are isomorphic 0 en Finite implies subnormal join property 0 en Finite normal implies amalgam-characteristic 0 en Finite not implies composition factor-permutable 0 en Finite not implies composition factor-unique 0 en Finitely many subgroups iff finite 0 en Finiteness is extension-closed 0 en Fitting subgroup is normal-isomorph-free in finite 0 en Frattini subgroup is nilpotent in finite 0 en Hall not implies automorph-conjugate 0 en Hall not implies order-isomorphic 0 en Hall not implies procharacteristic 0 en Hereditarily characteristic not implies cyclic in finite 0 en Homocyclic normal implies finite-pi-potentially fully invariant in finite 0 en Homocyclic normal implies potentially fully invariant in finite 0 en Isomorph-conjugacy is normalizer-closed in finite 0 en Monolith is fully invariant in co-Hopfian group 0 en Nilpotent Hall implies isomorph-conjugate 0 en Normal not implies finite-pi-potentially characteristic in finite 0 en Normal not implies normal-extensible automorphism-invariant in finite 0 en Periodic not implies locally finite 0 en Permutable implies subnormal in finite 0 en Pyber's theorem on logarithmic quotient of number of nilpotent groups to number of groups approaching unity 0 en Schur index of irreducible character in characteristic zero divides exponent 0 en Square-free implies solvability-forcing 0 en Subgroup of index equal to least prime divisor of group order is normal 0 en Sylow implies intermediately isomorph-conjugate 0 en Sylow implies order-dominating 0 en Sylow implies pronormal 0 en Sylow normalizer implies abnormal 0 en Sylow normalizer implies weakly abnormal 0 en Sylow of normal implies pronormal 0 en Sylow subgroups exist 0 en 3 Sylow-permutable implies subnormal in finite 0 en Upward-closed characteristic not implies cyclic-quotient in finite 0 en