On the Taketa Theorem Original Research Article

In this note we prove that a finite group is almost solvable if every irreducible character is induced from a character of degree at most 4 (more precisely, such a groupGis solvable, orG/S(G)congruent withA5, where S(G) is the solvable radical ofG). In particular, if every irreducible character ofGis induced from a character of degree at most 3 thenGis solvable. This result justifies Conjecture 3 from a previous paper by the author (Proc. Amer. Math. Soc.1231 (1995), 3263–3268). Our proofs use the fact that A5(congruent withPSL(2.5)) and PSL(2, 7) are the only complex linear nonabelian simple groups of degree at most 4.