Products of involutions in symplectic groups over general fields (I)

Abstract

Let s be an n-dimensional symplectic form over an arbitrary field with characteristic not 2, with n>2.The simplicity of the group Sp(s)/{±id} and the existence of a non-trivial involution in Sp(s) yield that every element of Sp(s) is a product of involutions.Extending and improving recent results of Awa, de La Cruz, Ellers and Villa with the help of a completely new method, we prove that if the underlying field is infinite, every element of Sp(s) is the product of four involutions if n is a multiple of 4, and of five involutions otherwise. The first part of this result is shown to be optimal for all multiples of 4 and all fields, and is shown to fail for the fields with three elements and for n=4. Whether the second part of the result is optimal remains an open question.Finite fields will be tackled in a subsequent article.