On a fixed point formula of Navarro–Rizo

Abstract:

Let $G$ be a $\pi$-separable group with a Hall $\pi$-subgroup $H$ or order $n$. For $x\in H$ let $\lambda (x)$ be the number of Hall $\pi$-subgroups of $G$ containing $x$. We show that $\prod _{d\mid n}\prod _{x\in H}\lambda (x^{d})^{\frac {n}{d}\mu (d)}=1$, where $\mu$ is the Möbius function. This generalizes fixed point formulas for coprime actions by Brauer, Wielandt and Navarro–Rizo. We further investigate an additive version of this formula.