On imprimitive multiplicity-free permutation groups the degree of which is the product of two distinct primes

Let P Q denote the set of n ∈ N such that n is a product of two primes with gcd ( n , φ ( n ) ) = 1 where φ is the Euler function. In this article we aim to find n ∈ P Q such that any imprimitive permutation group of degree n is multiplicity-free. Let R denote the set of such integers in P Q . Our main theorem shows that there are at most finitely many Fermat primes if and only if | P Q − R | is finite, whose proof is based on the classification of finite simple groups.