The primitivity of transitive permutation groups is examined in terms of conditions on a system of generators. The authors give a criterion of primitivity for the Galois group of an irreducible trinomial with integer coefficients of the form \(X^n+ ac^n X^s+ bc^n\) using the study of the inertia groups of primes in the splitting field of such trinomials over \(\mathbb{Q}\). Conditions under which the Galois groups over \(\mathbb{Q}\) of irreducible trinomials \(X^n+ ac^nX^s + bc^n\) are isomorphic to the full symmetric group \(S_n\) are obtained, improving earlier results of \textit{H. Osada} [J. Number Theory 25, 230-238 (1987; Zbl 0608.12010)].