Some sufficient conditions for a finite group to be supersolvable.
The authors extend the results of \textit{Ya. G. Berkovich} [Mat. Sb., N. Ser. 74(116), 75-92 (1967); translation in Math USSR, Sb. 3(1967), 69-83 (1969; Zbl 0183.02901)] and \textit{M. Asaad} [Commun. Algebra 38, No. 10, 3616-3620 (2010; Zbl 1205.20026)] by proving the following. Theorem~1.1. Let \(G\) be a group of odd order. Let \(G=G_1G_2\cdots G_n\) be the product of pairwise permutable subgroups \(G_1,G_2,\dots,G_n\). If the Sylow \(p\)-subgroups of all \(G_i\) are cyclic, then \(G\) is \(p\)-supersolvable. Theorem~1.3. Let \(G\) be a group of even order. Let \(G=G_1G_2\cdots G_n\) be the product of pairwise permutable subgroups \(G_1,G_2,\dots,G_n\) such that \(G_2G_3\cdots G_n\) is of odd order and \(G_1G_i\) is a mutually \(m\)-permutable product (\(i=2,3,\dots,n\)), that is, \(G_1\) permutes with every maximal subgroup of \(G_i\) and \(G_i\) permutes with every maximal subgroup of \(G_1\). If the Sylow subgroups of all \(G_i\) are cyclic, then \(G\) is supersolvable.
scientific article; zbMATH DE number 6121083