Finite groups having only one irreducible representation of degree greater than one

Abstract

In this paper all finite groups having exactly one irreducible K-representation of degree greater than one are determined, where K is an algebraically closed field of characteristic zero. The quaternion group of order eight and the dihedral group of order eight are nilpotent groups with this property, while the symmetric group on three letters and the alternating group on four letters are solvable although not nilpotent examples. The theorem below will show how typical the above examples really are. In the following all groups are finite. If G is a group, let GI denote the derived group of G and Z(G) the center of G. Let K be an algebraically closed field of characteristic zero.