A note on \(p\)-nilpotence and solvability of finite groups.
Let \(G\) be a finite group, \(p\) a prime number, \(P\) a \(p\)-Sylow subgroup. In recent papers, for instance, in \textit{A. Ballester-Bolinches} and \textit{X. Guo} [J. Algebra 228, No. 2, 491-496 (2000; Zbl 0961.20016)], characterisations were given for \(G\) to be \(p\)-nilpotent, solvable, supersolvable under certain hypotheses. The authors provide examples that show the necessity of these hypotheses. Further, certain results in these papers are extended by the authors, such as: if \(\Omega_1(P\cap O^p(G))\) is central in \(O^p(G)\) and \(P\) is quaternion-free in case \(p=2\) then the group \(G\) is nilpotent; moreover, under the same hypotheses for \(p=2\) and in case there is a nilpotent maximal subgroup then the group is solvable.
scientific article; zbMATH DE number 5549378
scientific article