Length parameters of finite groups and their Hall subgroups

Abstract

Let $\ddot{I}€$ be a set of primes containing $2$ and an odd prime $p$. It is proved that if a finite group $G$ has a Hall $\ddot{I}€$-subgroup $H$, then the non-$p$-soluble length of $G$ is bounded above by the generalized Fitting height of $H$. The proof uses the fact, obtained in [4] using the classification of finite simple groups, that a finite simple group of order divisible by $p$ cannot have a nilpotent Hall $\{2,p\}$-subgroup. As a corollary, it is proved that if in addition $H$ is soluble, then the non-$p$-soluble length of $G$ is bounded above by $2l_2(H)+1$, where $l_2(H)$ is the $2$-length of $H$.