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

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