On abelian quotients of primitive groups

Abstract

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