Commutators in the symplectic group
We show that if \(k\) is a field of characteristic different from 2 in which \(-1\) is a square, there is a conjugacy class \(K\) of elements of order 4 in the symplectic group \(Sp(2n,k)\) with the property that each element in the group is expressible as a product of two elements in \(K\). As a consequence, each element of \(Sp(2n,k)\) is a commutator if \(k\) satisfies this condition. By way of contrast, we show that if \(-1\) is not a sum of two squares in \(k\), the central involution \(-I\) is not a commutator in \(Sp(4n+2,k)\).
scientific article; zbMATH DE number 4021449