Abstract: In the theory of finite groups, the Thompson conjecture on the characterization of finite simple groups within the class of centerless groups by the set of sizes of their conjugacy classes is well-known. Recently, a positive answer to this
conjecture has been obtained. The next obvious step is to ask which finite groups are characterized within the class of centerless groups by the set of sizes of their conjugacy classes, without assuming they are simple. The talk plans to provide an
overview of the existing results on this topic.
