Recent zbMATH articles in MSC 20Hhttps://zbmath.org/atom/cc/20H2022-07-25T18:03:43.254055ZWerkzeugOn decompositions of matrices into products of commutators of involutionshttps://zbmath.org/1487.150152022-07-25T18:03:43.254055Z"Son, Tran Nam"https://zbmath.org/authors/?q=ai:son.tran-nam"Dung, Truong Huu"https://zbmath.org/authors/?q=ai:dung.truong-huu"Ha, Nguyen Thi Thai"https://zbmath.org/authors/?q=ai:ha.nguyen-thi-thai"Bien, Mai Hoang"https://zbmath.org/authors/?q=ai:mai-hoang-bien.Let \(\mathbb{F}\) be a field with at least three elements. Denote by \(\mathrm{SL}_n(\mathbb{F})\) the set of \(n \times n\) matrices with entries in \(\mathbb{F}\) and with determinant 1. A matrix \(A\) is an involution if \(A^2 = I_n\), where \(I_n\) is the \(n \times n\) identity matrix. A commutator of two involutions is of the form \(BCB^{-1}C^{-1}\), where \(B\) and \(C\) are involutions.
The main theorem of the paper is the following: If \(n\) is a natural greater than 1, then every matrix in \(\mathrm{SL}_n(\mathbb{F})\) is a product of at most two commutators of involutions.
Reviewer: Julio BenÃtez Lopez (Valencia)