Document Type: Ischia Group Theory 2018
Departamento de Matematicas, Universidad Jaume I, 12071, Castellon, Spain
Instituto Universitario de Matematica Pura y Aplicada, Universitat Politecnica de Valencia, 46022, Valencia, Spain
We summarize several results about non-simplicity, solvability and normal structure of finite groups related to the number of conjugacy classes appearing in the product or the power of conjugacy classes. We also collect some problems that have only been partially solved.