Wreath products and the non-coprime k ( GV ) problem

Let G = X H be the wreath product of a nontrivial finite group X with k conjugacy classes and a transitive permutation group H of degree n acting on the set of n direct factors of X n . If H is semiprimitive, then k(G) ≤ kⁿ for every sufficiently large n or k . This result solves a case of the non-coprime k ( GV ) problem and provides an affirmative answer to a question of Garzoni and Gill for semiprimitive permutation groups. The proof does not require the classification of finite simple groups.