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 M\"obius function. This generalizes fixed point formulas for coprime actions by Brauer, Wielandt and Navarro-Rizo. We further investigate an additive version of this formula.