For each finite group E, let O（E） be a binary relation on the set of all subgroups of E. If A and B are subgroups of a finite group G, then we say that the pair （A, B） enjoys the gradewise property （resp., genera...For each finite group E, let O（E） be a binary relation on the set of all subgroups of E. If A and B are subgroups of a finite group G, then we say that the pair （A, B） enjoys the gradewise property （resp., generalized gradewise property） in G if G has a normal series Г：1=G0≤G1≤Gt=Gsuch that for each i=1,...,t,we have （（A∩Gi）Gi-1/BGi-1/Gi-1）∈θ（G/Gi-1）（resp., we have （（A∩Gi）Gi-1/BGi-1/Gi-1）∈θ（G/Gi-1）Using these concepts, we obtain some new characterizations for solubility and supersolubility of finite groups and generalize some known results.展开更多
文摘For each finite group E, let O（E） be a binary relation on the set of all subgroups of E. If A and B are subgroups of a finite group G, then we say that the pair （A, B） enjoys the gradewise property （resp., generalized gradewise property） in G if G has a normal series Г：1=G0≤G1≤Gt=Gsuch that for each i=1,...,t,we have （（A∩Gi）Gi-1/BGi-1/Gi-1）∈θ（G/Gi-1）（resp., we have （（A∩Gi）Gi-1/BGi-1/Gi-1）∈θ（G/Gi-1）Using these concepts, we obtain some new characterizations for solubility and supersolubility of finite groups and generalize some known results.
