Fibers of the square map in finite groups of Lie type and an application

Abstract

Let $G$ be a finite classical group, specifically one of the general linear, unitary, symplectic, or orthogonal groups defined over a finite field of odd characteristic. For any element $g\in G$, we consider the fiber of the squaring map at $g$, which is the set of all elements $h\in G$ satisfying $h^2 = g$. In this work, we determine the cardinality of these fibers for arbitrary elements $g\in G$. Our analysis enables us to compute the total number of real conjugacy classes in $G$. The central tool for this is a recent resolution of Brauers Problem 14.