On imprimitive groups with small degree (Q1192077)

Let \(G\) be a transitive, imprimitive permutation group of degree \(d = p^ 2\) or \(pq\), where \(p\) and \(q\) are primes, \(p>q\), and let \(n\) be the number of systems of imprimitivity for \(G\) (i.e., the number of proper subgroups properly containing a fixed stabilizer subgroup). If \(n\geq 3\) then \(n=p+1\) and \(G\) is a subgroup of the permutation group \(\{x\mapsto ax+v\}\) of a 2-dimensional vector space \(V\) over \(GF(p)\), where \(a\in GF(p)^*\) and \(v\in V\) (if \(d = p^ 2\)), or \(G\) is a nonabelian, regular permutation group of order \(pq\) (if \(d=pq\) and \(q\) divides \(p- 1\)). Some information for the case \(n=2\) is also obtained. The proof makes use of the classification of transitive groups of prime degree.