Gradewise Properties of Subgroups and Their Applications

Abstract

For each finite group E, let Θ(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) Θ in G if G has a normal series Γ : 1 = G0 [Formula: see text] such that for each [Formula: see text], we have [Formula: see text] (resp., we have [Formula: see text]. Using these concepts, we obtain some new characterizations for solubility and supersolubility of finite groups and generalize some known results.