Mutually permutable products and conjugacy classes. (Q1955625)

Let \(G\) be a group, two subgroups \(H\) and \(K\) of \(G\) are `permutable' if the set \(HK\) is a subgroup of \(G\), they are `mutually permutable' if \(H\) permutes with every subgroup of \(K\) and \(K\) permutes with every subgroup of \(H\). Let the (finite) group \(G=G_1G_2\cdots G_r\) be the product of pairwise permutable subgroups \(G_1,G_2,\dots,G_r\). The aim of the paper under review is study how does the structure of the factors \(G_i\) (\(1\leq i\leq s\)) affect the structure of the whole group \(G\). In particular Theorem 1.1 states that: (1) no conjugacy class length \(|x^G|\), where \(x\) is a \(p\)-regular element of prime power order in \(\bigcup_{i=1}^rG_i\), is divisible by \(p\) if and only if \(G=O_p(G)\times O_{p'}(G)\); (2) \(|x^G|\) is not divisible by \(p\) for every element \(x\in\bigcup_{i=1}^rG_i\) if and only if \(G=O_p(G)\times O_{p'}(G)\) with \(O_p(G)\) Abelian.