On a fixed point formula of Navarro-Rizo

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.