Powers of commutators in linear algebraic groups (Q6622904)

\textit{K. Honda} [Comment. Math. Univ. St. Pauli 2, 9--12 (1953; Zbl 0052.02204)] proved that every finite group satisfies the following property: if \(\gamma\) is a commutator in \(G\) and \(\langle \gamma \rangle = \langle \delta \rangle\), then \(\delta\) is also a commutator. Although Honda's original proof relied on character theory, \textit{H. W. Lenstra jun.} [Oper. Res. Lett. 51, No. 1, 17--20 (2023; Zbl 1525.20029)] later proved the Honda property for finite groups, without using character theory.\N\NOn the contrary to the finite case, there exist infinite groups that do not satisfy the Honda property, as noted by \textit{S. J. Pride} [Trans. Am. Math. Soc. 234, 483--496 (1977; Zbl 0366.20022)].\N\NIn this elegant paper, the author uses deep model-theoretic arguments to demonstrate that linear algebraic groups over algebraically closed fields, pseudo-finite fields and valuation rings over non-Archimedean local fields satisfy the Honda property. He further extends this result to profinite groups.