Bounding the number of conjugacy classes of a permutation group

Abstract For a finite group G, let k(G) denote the number of conjugacy classes of G. If G is a finite permutation group of degree n > 2, then k(G) ≤ 3 (n −1)/2. This is an extension of a theorem of Kovács and Robinson and in turn of Riese and Schmid. If N is a normal subgroup of a completely reducible subgroup of GL(n, q), then k(N ) ≤ q5n . Similarly, if N is a normal subgroup of a primitive subgroup of Sn , then k (N ) ≤ p (n) where p (n) is the number of partitions of n. These bounds improve results of Liebeck and Pyber.