On Isaacs’ three character degrees theorem

Abstract

Isaacs has proved that a finite group G G is solvable whenever there are at most three characters of pairwise distinct degrees in Irr ⁡ ( G ) \operatorname {Irr}(G) (Isaacs’ three character degrees theorem). In this note, using Isaacs’ result and the classification of the finite simple groups, we prove the solvability of G G whenever Irr ⁡ ( G ) \operatorname {Irr}(G) contains at most three monolithic characters of pairwise distinct degrees. §2 contains some additional results about monolithic characters.