Products of conjugacy classes in finite unitary groups $GU(3,q^2)$ and $SU(3,q^2)$

TL;DR

This paper investigates the products of conjugacy classes in finite unitary and special unitary groups over finite fields, providing a complete classification of class products that do not include the identity element.

Contribution

It offers a comprehensive solution to the class product problem for GU(3,q^2) and SU(3,q^2), extending previous results to these specific groups.

Findings

Complete classification of conjugacy class products not containing the identity

Explicit descriptions of conjugacy class interactions in GU(3,q^2) and SU(3,q^2]

Advances understanding of product structures in finite unitary groups

Abstract

For the finite groups GU(3), SU(3), GL(3), SL(3) over a finite field we solve the class product problem, i.e., we give a complete list of $m$-tuples of conjugacy classes whose product does not contain the identity matrix.