On a fixed point formula of Navarro-Rizo (Q6583739)

Let \(G\) be a finite group of odd order and let \(A=\langle \alpha, \beta \rangle\) be a Klein group acting on \(G\). R. Brauer's fixed point formula states that\N\[\N|G| \cdot |C_{G}(A)|^{2}=|C_{G}(\alpha)| \cdot |C_{G}(\beta)| \cdot |C_{G}(\alpha \beta)|.\N\]\NThis formula was generalized by [\textit{H. Wielandt}, Math. Z. 73, 146--158 (1960; Zbl 0093.02302)] to any finite \(p\)-group \(A\) using Möbius functions on the lattice of cyclic subgroups of \(A\). Another generalization was provided by [\textit{G. Navarro} and \textit{N. Rizo}, Proc. Am. Math. Soc. 144, No. 10, 4199--4204 (2016; Zbl 1362.20020)].\N\NLet \(\pi \subseteq \pi(G)\) and assume that \(G\) is \(\pi\)-separable. For a \(\pi\)-element \(x\in G\) let \(\lambda_{G}(x)\) be the number of Hall \(\pi\)-subgroups of \(G\) containing \(x\). If \(H\) is a \(\pi\)-Hall subgroup of order \(n\) of \(G\), the author proves that\N\[\N\prod_{d \mid n} \left( \prod_{x \in H} \lambda_{G}(x^{d})^{\frac{n}{d}} \right)^{\mu(d)}=1\N\]\Nwhere \(\mu\) is the Möbius function. This generalizes fixed point formulas for coprime actions by Brauer, Wielandt [loc. cit.] and Navarro-Rizo [loc. cit.]. The author further investigates an additive version of this formula.