On nilpotent and solvable quotients of primitive groups

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 $n^β$ and any solvable quotient of $G$ has order at most $n^{α+1}$ where $β=\log 32/ \log 9$ and $α=(3 \log (48)+\log (24))/ (3 \cdot \log (9))$. This was motivated by a result of Aschbacher and Guralnick

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