# Conjugacy classes of centralizers in the group of upper triangular matrices

## TL;DR

This paper investigates the structure of conjugacy classes of centralizers in the group of upper triangular matrices, showing that the number of z-classes is infinite over infinite fields for matrices of size at least 6, and finite over finite fields.

## Contribution

It establishes the finiteness or infiniteness of z-classes in upper triangular matrix groups depending on the field's nature and matrix size.

## Key findings

- Number of z-classes is infinite over infinite fields for matrices of size ≥ 6.
- Number of z-classes is finite over finite fields.
- Provides classification criteria for z-classes in upper triangular matrix groups.

## Abstract

Let G be a group. Two elements x and y in G are said to be in the same z-class if their centralizers in G are conjugate within G. In this paper, we prove that the number of z-classes in the group of upper triangular matrices is infinite provided that the field is infinite and size of the matrices is at least 6, and finite otherwise.