A nilpotency criterion for some verbal subgroups

TL;DR

The paper proves a sharp nilpotency criterion for verbal subgroups $w(G)$ generated by simple commutator words $w=[x_{i_1},\dots,x_{i_k}]$ with $i_1\neq i_j$ for all $j>1$: in finite groups, $w(G)$ is nilpotent exactly when $|ab|=|a||b|$ for all coprime-order $w$-values $a,b$ in $G_w$. The proof first handles the soluble case using Fitting height and tower methods to force nilpotency, and then treats the general case by reducing to minimal simple groups (via Thompson’s classification) to obtain solubility and apply the soluble argument. The results extend to residually finite groups provided $G_w$ is finite, with corollaries on finiteness and virtual nilpotency; the paper also includes explicit counterexamples showing the necessity of the hypotheses for words beyond the specified simple-commutator form.

Abstract

The word $w=[x_{i_1},x_{i_2},\dots,x_{i_k}]$ is a simple commutator word if $k\geq 2, i_1\neq i_2$ and $i_j\in \{1,\dots,m\}$, for some $m>1$. For a finite group $G$, we prove that if $i_{1} \neq i_j$ for every $j\neq 1$, then the verbal subgroup corresponding to $w$ is nilpotent if and only if $|ab|=|a||b|$ for any $w$-values $a,b\in G$ of coprime orders. We also extend the result to a residually finite group $G$, provided that the set of all $w$-values in $G$ is finite.

Theorems & Definitions (27)

• Theorem 1.1
• Corollary 1.2
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• proof
• Lemma 2.5
• proof
• Proposition 2.6