Real and strongly real classes in finite linear groups

TL;DR

This paper classifies real and strongly real conjugacy classes in various finite linear groups, providing formulas for their counts, which advances understanding of their algebraic structure.

Contribution

It offers a comprehensive classification and explicit formulas for real and strongly real classes across multiple finite linear groups, including covers of PSL_n(q).

Findings

Classified real and strongly real conjugacy classes in several finite linear groups.

Derived formulas for counting these classes in each group.

Enhanced understanding of the structure of finite linear groups.

Abstract

We classify the real and strongly real conjugacy classes in $GL_n(q)$, $SL_n(q)$, $PGL_n(q)$, $PSL_n(q)$, and all quasi-simple covers of $PSL_n(q)$. In each case we give a formula for the number of real, and the number of strongly real, conjugacy classes.