The Amit-Ashurst conjecture for finite metacyclic \(p\)-groups (Q6174105)

Let \(w\) be a word in \(k\) variables. The \textit{probability distribution} \(P_{w,G}\) induced by \(w\) on the group \(G\) is defined as \[ P_{w,G}(g)=\frac{|\{(g_1,\ldots,g_k)\in G^k\,:\, w(g_1,\ldots,g_k)=g\}|}{|G|^k} \] for every \(g\in G\). The \textit{Amit-Ashurst conjecture} states that if \(G\) is finite and nilpotent, then \[ P_{w,G}(g)\geq|G|^{-1} \] for all \(g\) in the set of words values of \(w\) in \(G\). The aim of the paper under review is to show that the conjecture holds for finite \(p\)-groups with a cyclic maximal subgroup. The proof of this result is broken into two parts. First, they deal with some larger family of metacyclic \(p\)-groups, and then they move to the general case using the classification of finite \(p\)-groups with a cyclic maximal subgroup.