The number of Hall $\pi$-subgroups of a $\pi$-separable group

Abstract:

We observe a simple formula to compute the number $\nu _\pi (G)$ of Hall $\pi$-subgroups of a $\pi$-separable finite group $G$ in terms of only the action of a fixed Hall $\pi$-subgroup of $G$ on a set of normal $\pi ’$-sections of $G$. As a consequence, we obtain that $\nu _\pi (K)$ divides $\nu _\pi (G)$ whenever $K$ is a subgroup of a finite $\pi$-separable group $G$. This generalizes a recent result of Navarro. In addition, our method gives an alternative proof of Navarro’s result.