Let \(w \in F_{k}\) be a word in the free group of rank \(k\). For any group \(G\) and arbitrary \(g_1, \ldots, g_k \in G\), elements of the form \(w(g_1, \ldots , g_k)\) are called \(w\)-values in \(G\). The set of all \(w\) values is denoted by \(w(G)\) and \(G(w)=\langle G_{w} \rangle\) is the verbal subgroup of \(G\) corresponding to \(w\).
If \(G\) is a group and \(p\) a prime, then \(G\) satisfies \(P(w, p)\) if the prime \(p\) divides \(o(xy)\), for every \(x \in G_{w}\) of \(p'\)-order and for every non-trivial \(y \in G_{w}\) of order divisible by \(p\).
Let \(\gamma_{k}\) be the word defined inductively as \(\gamma_{1}=x_{1}\) and \(\gamma_{k}=[\gamma_{k-1},x_{k}]\) if \(k \geq 2\). The first result proven in this paper is (Theorem C): Let \(G\) be a finite group. Then \(\gamma_{k}(G)\) is \(p\)-nilpotent if and only if \(G\) satisfies \(P(\gamma_{k},p)\).
Let \(\delta_{k}(x_{1},\ldots, x_{2^{k}}) \in F_{2^{k}}\) be the word defined inductively as \(\delta_{0}=x_{1}\) and \(\delta_{k}=[\delta_{k-1}(x_{1},\ldots,x_{2^{k-1}}),\delta_{k-1}(x_{2^{k-1}+1},\ldots,x_{2^{k}})]\). It is clear that \(\delta_{0}=\gamma_{1}\) and \(\delta_{1}=\gamma_{2}\) and therefore it is sufficient to consider the cases in which it is \(k \geq 2\).
Theorem D: Let \(G\) be a finite soluble group and \(k\in \mathbb{N}\), \(k \geq 2\). Then \(G^{(k)}\) is \(p\)-nilpotent if and only \(G\) satisfies \(P(\delta_{k},p)\).