### Character degrees and conjugacy class sizes of finite groups

The concept of a group was invented in the midst of attempts to solve polynomial equations, dating back to the work of Lagrange, Abel, Galois, and others at the beginning of the 19th century. Group representation theory and character theory were originally developed as a way of analyzing groups in terms of linear transformations or matrices. We propose to exploit several interrelationships between the structure of a finite group, the degrees of its irreducible complex representations (or characters), and the sizes of its conjugacy classes.

We hope to prove that complex group algebras, character degrees, and conjugacy class sizes are enough to obtain structural results of certain groups, especially finite simple groups and more generally finite quasisimple groups. This builds upon a long-standing problem of Brauer about the determination of non-isomorphic groups having isomorphic complex group algebras and a conjecture of Huppert that finite nonabelian simple groups are essentially determined by the set of their character degrees, a result that does not extend to solvable groups or groups in general.