Conjugacy classes of centralizers in unitary groups

Abstract

Let G be a group. Two elements x,y∈G are said to be in the same z-class if their centralizers in G are conjugate within G. Consider F a perfect field of characteristic ≠2, which has a non-trivial Galois automorphism of order 2. Further, suppose that the fixed field F0 has the property that it has only finitely many field extensions of any finite degree. In this paper, we prove that the number of z-classes in the unitary group over such fields is finite. Further, we count the number of z-classes in the finite unitary group Un(q), and prove that this number is the same as that of GLn(q) when q>n.