We show that for any $n$ and $q$, the number of real conjugacy classes in $\mathrm{ PGL}(n, \mathbb{ F}_q)$ is equal to the number of real conjugacy classes of $\mathrm{ GL}(n, \mathbb{ F}_q)$ which are contained in $\mathrm{ SL}(n, \mathbb{ F}_q)$, refining a result of Lehrer, and extending the result of Gill and Singh that this holds when $n$ is odd or $q$ is even. Further, we show that this quantity is equal to the number of real conjugacy classes in $\mathrm{ PGU}(n, \mathbb{ F}_q)$, and equal to the number of real conjugacy classes of $\mathrm{ U}(n, \mathbb{ F}_q)$ which are contained in $\mathrm{ SU}(n, \mathbb{ F}_q)$, refining results of Gow and Macdonald. We also give a generating function for this common quantity.
Categories: math.GR math.CO