ON THE TAKETA THEOREM

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 group G is solvable, or G/S(G) congruent to A( 5), where S(G) is the solvable radical of G). In particular, if every irreducible character of G is induced from a character of degree at mo st 3 then G is solvable. This result justifies Conjecture 3 from a pre vious paper by the author (Proc. Amer. Math. Sec. 123 1 (1995), 3263-3 268). Our proofs use the fact that A(5) (congruent to PSL(2.5)) and PS L(2,7) are the only complex linear nonabelian simple groups of degree at most 4. (C) 1996 Academic Press, Inc.