On abelian quotients of primitive groups

Abstract:

It is shown that if $G$ is a primitive permutation group on a set of size $n$, then any abelian quotient of $G$ has order at most $n$. This was motivated by a question in Galois theory. The field theoretic interpretation of the result is that if $M/K$ is a minimal extension and $L/K$ is an abelian extension contained in the normal closure of $M$, then the degree of $L/K$ is at most the degree of $M/K$.