On numbers which are orders of nilpotent groups with bounded class (Q6620114)

Let \(m \geq 2\) be an integer and let \(m=p_{1}^{\alpha_{1}}p_{2}^{\alpha_{2}} \ldots p_{r}^{\alpha_{r}}\) be its prime factorization. A result by \textit{T. W. Müller} [J. Algebra 300, No. 1, 10--15 (2006; Zbl 1099.20014)] states that every group of order \(m\) is nilpotent of class at most \(n\) if and only if (a) \(p_{i}\) does not divide \(p_{j}^{k} -1\), for all \(i \not =j\) and all \(k \in [1,\dots, \alpha_{j}]\), (b) \(1 \leq \alpha_i \leq n+1\), for all \(i\).\N\NThe author characterizes those numbers \(m\) for which any group of order \(m\) is an \(n\)-Engel group and those numbers \(m\) for which any group of order \(m\) has all its subgroups subnormal of defect at most \(n\) (they are essentially the same that satisfy conditions (a) and (b) described above).