On groups with large verbal quotients (Q6634496)

Let \(F=\langle x_{1}, \ldots , x_{n} \rangle\) be a free group of rank \(n\) and let \(w=w(x_{1}, \ldots, x_{n}) \in F\) be a word. The verbal subgroup \(w(G)\) of a group \(G\) is the subgroup \(\langle w(g_{1}, \ldots, g_{n}) \mid g_{1}, \ldots, g_{n} \in G \rangle\) generated by the set of all \(w\)-values in \(G\).\N\N\textit{J. G. Thompson} [J. Algebra 13, 149--151 (1969; Zbl 0194.03902)] observed that, if \(G\) is a finite \(p\)-group (\(p\) a prime) such that \(|H : H'| < |G: G'|\) for every \(H < G\), then the nilpotency class of \(G\) is at most \(2\). The second author [Proc. Am. Math. Soc. 150, No. 8, 3241--3244 (2022; Zbl 1506.20039)] (Lemma 2), remarked that the previous strict inequality implies nilpotency and so the hypothesis that \(G\) is a \(p\)-group can be removed. By analogy, a group \(G\) is called \(w\)-maximal if \(|H: w(H)| < |G:w(G)|\) for every \(H < G\) (this concept was introduced by \textit{J. González-Sánchez} and \textit{B. Klopsch} [J. Algebra 328, No. 1, 155--166 (2011; Zbl 1223.20008)]) and weakly \(w\)-maximal if \(|H: w(H)| \leq |G:w(G)|\) for every \(H \leq G\).\N\NIn the paper under review, the authors prove the following:\N\NTheorem 1.2. Let \(n= 2, 3\). Every weakly \(\gamma_{n}\)-maximal finite group is the direct product of weakly \(n\)-maximal finite \(p\)-groups. They then show by an example that Theorem 1.2 is the best possible because it fails for each \(n \geq 4\).\N\NAnother relevant result is Theorem 1.3: The properties of being weakly \(w\)-maximal and \(w\)-maximal are invariant under \(w\)-isologism.\N\NThe paper contains many other interesting insights, in particular the reviewer points out three questions at the end of the paper.