A generalization of Kramer's theory (Q783240)

Let \(\sigma\) be a partition of the set of all prime numbers, and for each positive integer \(n\) let \(\sigma(n)\) denote the set of all elements of \(\sigma\) that contain some divisor of \(n\). Then the positive integers \(m\) and \(n\) are called \textit{\(\sigma\)-coprime} if \(\sigma(m)\cap\sigma(n)=\emptyset\). A group class \(\mathcal{F}\) (consisting only of finite groups) is said to be \textit{\(\Gamma_t^\sigma\)-closed} for some integer \(t>1\) if \(\mathcal{F}\) contains every finite group \(G\) admitting \(t\) subgroups \(A_1,\ldots,A_t\) such that \(G=A_iA_j\) whenever \(i\neq j\) and the indices \(|G:N_G(A_1)|,\ldots,|G:N_G(A_t)|\) are pairwise \(\sigma\)-coprime.\par In the paper under review, the authors study the above concepts, and in particular they prove that certain formations of generalized metanilpotent groups (related to the partition \(\sigma\)) are \(\Gamma_4^\sigma\)-closed.