Coprime commutators in finite groups

TL;DR

The paper investigates when the coprime commutator subgroups $\gamma^*_k(G)$ and $\delta^*_k(G)$ of a finite group $G$ are nilpotent, linking this to precise order conditions on coprime commutators and their powers. It proves two main theorems: (A) $\gamma^*_k(G)$ is nilpotent if and only if $|ab|=|a||b|$ for $\gamma^*_k$-commutators $a,b$ of coprime orders; (B) $\delta^*_k(G)$ is nilpotent if and only if $|ab|=|a||b|$ for powers of $\delta^*_k$-commutators with coprime orders. The proofs combine reductions to the nilpotent residual $\gamma_{\infty}(G)$, detailed structural analysis of radicals and minimal simple groups, and coprime-action techniques to derive contradictions in minimal counterexamples. Consequences include characterizations of when $\gamma_{\infty}(G)$ is nilpotent and bounds on the Fitting height, with broader implications for solubility and the behavior of simple-commutator words. The results advance Burnside-type criteria for nilpotency in the setting of coprime commutators and provide a framework for extending order-based nilpotency criteria to other group-words.

Abstract

Let $G$ be a finite group and let $k \geq 2$. We prove that the coprime subgroup $γ_k^*(G)$ is nilpotent if and only if $|xy|=|x||y|$ for any $γ_k^*$-commutators $x,y \in G$ of coprime orders (Theorem A). Moreover, we show that the coprime subgroup $δ_k^*(G)$ is nilpotent if and only if $|ab|=|a||b|$ for any powers of $δ_k^*$-commutators $a,b\in G$ of coprime orders (Theorem B).

Theorems & Definitions (34)

• Theorem 1: baubau
• Theorem 2: BS
• Theorem 4: MT
• Theorem 5: bms
• Conjecture 6
• Lemma 8: pavel, Lemma 2.4
• Lemma 9: pavel, Lemma 2.6
• Theorem 10: pavel, Theorem 2.7
• Lemma 11
• Theorem 12: pavel, Theorem 2.5