Upper bounds on the number of conjugacy classes in unitriangular groups

TL;DR

This paper establishes new upper bounds on the number of conjugacy classes in the unitriangular group over finite fields and its lower central series groups, advancing understanding of their algebraic structure.

Contribution

It introduces novel upper bounds for conjugacy classes in $U_n(q)$ and its lower central series groups, improving previous estimates.

Findings

New upper bounds for conjugacy classes in $U_n(q)$

Upper bounds for groups in the lower central series of $U_n(q)$

Enhanced understanding of algebraic structure of unitriangular groups

Abstract

We provide a new upper bound on the number of conjugacy classes in the group $U_n(q)$ of unitriangular matrices over a finite field. We also compute a similar upper bound for every group in the lower central series of $U_n(q)$.