Groups with Representations of Bounded Degree

Let G be a discrete group with group algebra C G over the complex numbers C . In (5) Kaplansky essentially proves that if G has a normal abelian subgroup of finite index n , then all irreducible representations of C G have degree ≤ n . Our main theorem is a converse of Kaplansky's result. In fact we show that if all irreducible representations of C G have degree ≤ n , then G has an abelian subgroup of index not greater than some function of n . (The degree of a representation of C G for arbitrary G is defined precisely in § 3.)