On the composition length of finite primitive linear groups

Let $G$ be a finite primitive linear group over a field $K$, where $K$ is a finite field or a field of numbers. We bound the composition length of $G$ in terms of the dimension of the underlying vector space and of the degree of $K$ over its prime subfield. As a by-product, we prove a result of number theory which bounds thenumber of prime factors (counting moltiplicities), of $q^{n}-1$, where $q,n>1$ are integers, improving a result of Turull and Zame.