Maximal Non-commuting Sets in Certain Unipotent Upper-triangular Linear Groups

TL;DR

This paper determines the exact size of maximal non-commuting sets in the unipotent upper-triangular linear group over finite fields, using geometric structures to analyze their properties.

Contribution

It provides the first exact calculation of maximal non-commuting set sizes in $UU_4(_q)$ and introduces geometric methods for their analysis.

Findings

Exact size of maximal non-commuting sets in $UU_4(_q)$

Bounds on non-commuting set sizes using geometric structures

Explicit constructions of non-commuting sets

Abstract

We find the exact size of a maximal non-commuting set in unipotent uppertriangular linear group $UU_4(\mathbb{F}_q)$ in terms of a non-commuting geometric structure (Refer Definition [10]), where $\mathbb{F}_q$ is the finite field with $q$ elements. Then we get bounds on the size of such a set by explicitly finding certain non-commuting sets in the non-commuting structure.