# $p$-Power conjugacy classes in $U(n,q)$ and $T(n,q)$

## TL;DR

This paper investigates the behavior of p-power maps on the unitriangular and triangular groups over finite fields, revealing large conjugacy classes in the images and providing recursive formulas for counting elements.

## Contribution

It introduces new results on the structure of p-power maps in $U(n,q)$ and $T(n,q)$, including size estimates and recursive counting methods.

## Key findings

- The image of p-power maps contains large conjugacy classes.
- Recursive formulas are provided for counting elements in the image.
- The results extend understanding of p-power maps in algebraic groups.

## Abstract

Let $q$ be a $p$-power where $p$ is a fixed prime. In this paper, we look at the $p$-power maps on unitriangular group $U(n,q)$ and triangular group $T(n,q)$. In the spirit of Borel dominance theorem for algebraic groups, we show that the image of this map contains large size conjugacy classes. For the triangular group we give a recursive formula to count the image size.