Powers of commutators in linear algebraic groups

Abstract Let 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 non-archimedean local field. Let G= G (k). We prove that if G such that γ is a commutator and G such that = then δ is a commutator. This generalises a result of Honda for finite groups. Our proof uses the Lefschetz principle from first-order model theory.