$p$-nilpotency criteria for some verbal subgroups
Yerko Contreras Rojas, Valentina Grazian, Carmine Monetta
TL;DR
The authors introduce the Property $P(w,p)$ to relate the orders of $w$-values in a finite group to $p$-nilpotency, and they establish sharp verbal criteria for key groups-theoretic layers. They prove that for any $k\ge 2$, the $k$th term of the lower central series $\gamma_k(G)$ is $p$-nilpotent if and only if $G$ satisfies $P(\gamma_k,p)$. In the soluble case, the $k$th derived term $G^{(k)}$ is $p$-nilpotent if and only if $G$ satisfies $P(\delta_k,p)$, with $\delta_k$ being derived-words. The results extend known criteria for $p$-nilpotency to verbal subgroups generated by multilinear commutator words and provide a unifying framework based on $P(w,p)$ for analyzing $p$-nilpotency via word-values.
Abstract
Let $G$ be a finite group, let $p$ be a prime and let $w$ be a group-word. We say that $G$ satisfies $P(w,p)$ if the prime $p$ divides the order of $xy$ for every $w$-value $x$ in $G$ of $p'$-order and for every non-trivial $w$-value $y$ in $G$ of order divisible by $p$. If $k \geq 2$, we prove that the $k$th term of the lower central series of $G$ is $p$-nilpotent if and only if $G$ satisfies $P(γ_k,p)$. In addition, if $G$ is soluble, we show that the $k$th term of the derived series of $G$ is $p$-nilpotent if and only if $G$ satisfies $P(δ_k,p)$.