The influence of conjugacy class sizes on the structure of finite groups: a survey. (Q2892236)

There is a long line of research which is devoted to study the structural properties of finite groups by restrictions on the conjugacy classes. This brief sentence hides a large part of group theory, representation theory and combinatorics behind ``by restrictions on the conjugacy classes''. The authors of the present survey describe in fact some sophisticated techniques of investigation in order to study the influence of the conjugacy class sizes on the structure of finite groups. Among these techniques, it is possible to discover arithmetical methods, elements of graph theory, applications of the theory of P. Hall of solvable groups, classical methods of \(p\)-group theory and interesting applications of character theory and graph theory. The extensive (and recent) list of references shows the motivation to write a similar survey in the area. Among other things, there are recent conjectures and open questions.