Conjugacy classes of small sizes in the linear and unitary groups

Abstract

Abstract. Using the classical results of G. E. Wall on the parametrization and sizes of (conjugacy) classes of finite classical groups, we present some gap results for the class sizes of the general linear groups and general unitary groups as well as their variations. In particular, we identify the classes in of size up to and classes in of size up to . We then apply these gap results to obtain some bounds and limits concerning the zeta-type function encoding the conjugacy class sizes of these groups.