The number of Hall 𝜋-subgroups of a 𝜋-separable group

Abstract

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