Powers of commutators in linear algebraic groups

TL;DR

This paper proves that in certain linear algebraic groups over various fields, elements generating the same cyclic subgroup as a commutator are themselves commutators, extending Honda's finite group result using model theory.

Contribution

It generalizes Honda's finite group result to linear algebraic groups over algebraically closed, pseudo-finite, or valuation rings fields, employing the Lefschetz Principle.

Findings

Elements with the same cyclic subgroup as a commutator are also commutators.

The result applies to groups over algebraically closed, pseudo-finite, and valuation ring fields.

Uses model theory to extend finite group properties to algebraic groups.

Abstract

Let ${\mathscr G}$ be a linear algebraic group over $k$, where $k$ is an algebraically closed field, a pseudo-finite field or the valuation ring of a nonarchimedean local field. Let $G= {\mathscr G}(k)$. We prove that if $\gamma, \delta\in G$ such that $\gamma$ is a commutator and $\langle \delta\rangle= \langle \gamma\rangle$ then $\delta$ is a commutator. This generalises a result of Honda for finite groups. Our proof uses the Lefschetz Principle from first-order model theory.