On nilpotent and solvable quotients of primitive groups

Abstract

It is shown that if $G$ is a primitive permutation group on a set of size $n$, then any nilpotent quotient of $G$ has order at most $n^{\beta }$ and any solvable quotient of $G$ has order at most $n^{\alpha +1}$ where $\beta =\log 32/\log 9$ and $\alpha =(3\log (48)+\log (24))/(3\cdot \log (9))$. This was motivated by a result of Aschbacher and Guralnick