Products of symmetries in unitary groups

Abstract

Given a regular −-hermitian form on a finite-dimensional vector space V over a commutative field K of characteristic ≠2 such that the norm on K is surjective onto the fixed field of − (this is true whenever K is finite). Call an element σ of the unitary group a symmetry if σ 2 = 1 and the negative space of σ is 1-dimensional. If π is unitary and det π ∈ {1, − 1}, we prove that π is a product of symmetries (with a few exceptions when K = GF9 and dim V = 2) and we find the minimal number of factors in such a product.