On a fixed point formula of Navarro–Rizo

Let G G be a π π -separable group with a Hall π π -subgroup H H or order n n . For x ∈ H x∈ H let λ ( x ) λ (x) be the number of Hall π π -subgroups of G G containing x x . We show that ∏ d ∣ n ∏ x ∈ H λ ( x d ) n d μ ( d ) = 1 ∏ d n∏ x∈ Hλ (xᵈ)n/dμ (d)=1 , where μ μ 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.