Conjugacy classes of centralizers in unitary groups

TL;DR

This paper investigates the classification of elements in unitary groups based on conjugate centralizers over certain fields, establishing finiteness of such classes and comparing counts with general linear groups.

Contribution

It proves the finiteness of $z$-classes in unitary groups over specific fields and shows their count matches that of $GL_n(q)$ when $q>n$, providing new insights into group structure.

Findings

Number of $z$-classes in unitary groups is finite.

Count of $z$-classes in $U_n(q)$ equals that in $GL_n(q)$ for $q>n$.

Finiteness results depend on properties of the base field.

Abstract

Let $G$ be a group. Two elements $x,y \in G$ are said to be in the same $z$-class if their centralizers in $G$ are conjugate within $G$. Consider $\mathbb F$ a perfect field of characteristic $\neq 2$, which has a non-trivial Galois automorphism of order $2$. Further, suppose that the fixed field $\mathbb F_0$ 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 $U_n(q)$, and prove that this number is same as that of $GL_n(q)$ when $q>n$.