Conjugacy classes of centralizers in unitary groups

Abstract

Abstract Let G be a group. Two elements x , y ∈ G {x,y\in G} are said to be in the same z-class if their centralizers in G are conjugate within G. Consider 𝔽 {\mathbb{F}} a perfect field of characteristic ≠ 2 {\neq 2} , which has a non-trivial Galois automorphism of order 2. Further, suppose that the fixed field 𝔽 0 {\mathbb{F}_{0}} 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 U n ⁢ ( q ) {{\mathrm{U}}_{n}(q)} , and prove that this number is the same as that of GL n ⁢ ( q ) {{\mathrm{GL}}_{n}(q)} when q > n {q>n} .