A Gap of Three Between the Derived Lengths of a Finite Soluble Group and of a Maximal Subgroup (Kourovka Notebook Problem 17.91)

Problem 17.91 of the Kourovka Notebook, posed by V. S. Monakhov and also Question 9 of a survey of Monakhov and Trofimuk (2010), asks whether \(d(G)-d(M)\le k\) for an absolute constant \(k\), for all finite soluble groups \(G\) and maximal subgroups \(M\), where \(d\) is the derived length, and asks for the least such \(k\). We answer neither part. We show that \(k\ge 3\) if \(k\) exists: the split extension \(G=3^{1+2}_+\rtimes \mathrm{GL}(2,3)\) of order \(1296\) has derived length \(6\), and its maximal subgroup \(3^{1+2}_+\rtimes B\) of index \(4\), with \(B\) a Borel subgroup, has derived length \(3\). By an exhaustive computer search in the SmallGroups library, reduced by elementary lemmas, 1296 is the least order of a finite soluble group with a maximal subgroup \(M\) such that \(d(G)-d(M)\ge 3\), and exactly three groups of order 1296 have one. We prove that \(d(G)-d(M)\le 1+\min\{e,d(M_G)\}\), where \(M_G\) is the core of \(M\) and \(e=d(M/M_G)\), and deduce that a difference \(4\) needs \(|G|\ge 5184\); we also give a formula for split extensions which explains why the construction does not iterate in the simplest cases. In a second pair with difference 3, of order 41472, the group is a primitive permutation group of degree 64. The group \(G\) itself is a term of an extraspecial tower of Glasby and Howlett. Related results of Doerk, of Gol'chuk and Monakhov, and of Murashka and Vasil'ev concern other lengths or special cases and do not answer the question. The contribution is modest. This is an unrefereed note.