Primitive implies innately transitive

In the language of group actions
Suppose $$G$$ is a finite group with a primitive group action with a faithful primitive group action $$\alpha$$ on a set $$S$$. Then, there exists a minimal normal subgroup $$N$$ of $$G$$ such that the restriction of $$\alpha$$ to $$N$$ is still transitive to $$S$$.

In fact, the above statement holds for any minimal normal subgroup (and in fact, any nontrivial normal subgroup) of $$G$$. (This is the stronger fact that any primitive group is quasiprimitive).

In the language of group theory
Suppose $$M$$ is a core-free maximal subgroup of a group $$G$$. Then, $$G$$ has a minimal normal subgroup $$N$$ such that $$NM = G$$.

Related facts

 * Primitive implies quasiprimitive, Quasiprimitive implies innately transitive
 * Abelian minimal normal subgroup and core-free maximal subgroup are permutable complements
 * Abelian permutable complement to core-free subgroup is self-centralizing
 * Plinth theorem
 * Primitive implies Fitting-free or elementary abelian Fitting subgroup
 * Primitive solvable group acts on a set iff the set has prime power size

In the language of group theory
Since the action of $$G$$ is primitive, the stabilizer of any point is a core-free maximal subgroup. Call this $$M$$. In particular, $$M$$ does not contain any nontrivial normal subgroup. Hence, its product with any nontrivial normal subgroup $$N$$ is strictly bigger than $$M$$, so $$MN = G$$.