18.110 (2014)

The non-$p$-soluble length of a finite group $G$ is the number of non-$p$-soluble factors in a shortest normal series each of whose factors either is $p$-soluble or is a direct product of non-abelian simple groups of order divisible by $p$. For a given prime $p$ and a given proper group variety $\mathfrak{V}$, is there a bound for the non-$p$-soluble length of finite groups whose Sylow $p$-subgroups belong to $\mathfrak{V}$?