Gradewise properties of subgroups of finite groups.
Let \(G\) be a group, \(A\) a subgroup of \(G\) and let \(\theta\) be a group theoretic property. The subgroup \(A\) enjoys the gradewise property \(\theta\) in \(G\) whenever \(G\) has a normal series \(1=G_0<G_1<\cdots<G_t=G\) such that \((A\cap G_i)G_{i-1}/G_{i-1}\) enjoys property \(\theta\) in \(G_i/G_{i-1}\) for each \(i=1,2,\ldots,t\). In this paper the authors obtain some new characterizations of finite supersolvable and solvable groups. A typical result, among others, is Theorem A: A finite group \(G\) is solvable if and only if some maximal subgroup \(M\) of \(G\) is gradewise nilpotent and the Sylow \(2\)-subgroup of \(M\) is gradewise abelian in \(G\).
scientific article; zbMATH DE number 6485693