On a fixed point formula of Navarro-Rizo

TL;DR

The paper generalizes fixed-point formulas for coprime actions to $\pi$-separable groups with a Hall $\pi$-subgroup of order $n$, introducing $\lambda_G(x)$ as the number of Hall $\pi$-subgroups containing $x$. The main result is a multiplicative fixed-point formula: $ $\prod_{d\,\mid\, n}\Bigl(\prod_{x\in H}\lambda_G(x^{d})^{\frac{n}{d}}\Bigr)^{\mu(d)}=1$, extending formulas of Brauer, Wielandt and Navarro–Rizo. An additive analogue is established via $\beta_G(H):=\frac{1}{n^2}\sum_{h\in H}\sum_{d\,\mid\, n}\mu(d)\lambda_G(h^d)^{\frac{n}{d}}$, shown to be a non-negative integer and to vanish precisely when $1\ne H\unlhd G$; the approach uses permutation-characters and Möbius inversion on cyclic Hall subgroups, together with Wielandt-type fixed-point arguments. The results illuminate fixed-point phenomena for Hall subgroups in a broad class of finite groups and connect to classical distribution formulas for coprime actions.

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 $\prod_{d\mid n}\prod_{x\in H}λ(x^{d})^{\frac{n}{d}μ(d)}=1$, 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.

Theorems & Definitions (9)

• Theorem 1: Navarro--Rizo
• Theorem 2
• Theorem 3
• Proposition 4
• proof
• Theorem 5: Wielandt
• proof
• Theorem 6
• proof