On factorized finite groups in which certain subgroups of the factors permute (Q1273222)

Let the finite group \(G=G_1G_2\cdots G_r\) be the product of pairwise permutable subgroups \(G_i\). The author studies such factorized groups in which certain subgroups of distinct factors permute. If each \(G_i\) is \(p\)-soluble for some prime \(p\) and \(G_i\) permutes with every Sylow subgroup of every \(G_j\) for all \(j\), then \(G\) is \(p\)-soluble. If \(G\) is simple and each \(G_i\) is non-trivial and permutes with every Sylow subgroup of \(G_j\) for all \(j\), then \(G=G_1=G_2=\cdots=G_r\).