Bounding the Number of Conjugacy Classes of a Permutation Group

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) ≤ q. 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 improve results of Liebeck