Permutable products of supersoluble groups.
Let the finite group \(G=AB\) be the product of two supersoluble subgroups \(A\) and \(B\). It is well-known that \(G\) need not to be supersoluble, even if \(A\) and \(B\) are normal in \(G\). The authors consider the case that the subgroup \(A\) commutes with every subgroup of \(B\) and \(B\) commutes with every subgroup of \(A\), i.e. that \(G\) is the permutable product of \(A\) and \(B\). They show that in this case the Fitting factor group \(G/F(G)\) must be supersoluble and metabelian; also \(G/\text{Core}_G(A\cap B)\) is always supersoluble. This also provides a new proof of a theorem of Baer and a result of Asaad and Shaalan by which a mutually product of two supersoluble subgroups is supersoluble, provided that its commutator subgroup is nilpotent.
scientific article; zbMATH DE number 2116868