Products of commutators of dilatations
Let \(SL_nF\) denote the group of all \(n\times n\) matrices with determinant 1 over a field \(F\). Let \(\text{res }A\) denote \(\text{rank}(A-I)\) for \(A\in GL_nF\). A matrix \(A\in GL_nF\) is called a dilation if \(A\) is similar to a matrix \(I_{n-1} \oplus a\) (\(a\neq 1\)). In this paper, it is proved that if \(|F|>4\), then every matrix \(A\in SL_nF\) is the product of at most \([\text{res }A/2]+1\) commutators of dilatations for \(n\geq 2\) ([\ ] denotes the integer part of a rational number).
scientific article; zbMATH DE number 1331315
scientific article