Geodesic
Character Degrees and Derived Length of a Solvable Group
Canadian journal of mathematics, Tome 27 (1975) no. 1, pp. 146-151

Voir la notice de l'article provenant de la source Cambridge University Press

Let G be a finite group. (All groups considered here are finite). There exist several results which control the structure of G in terms of cd(G), the set of degrees of the irreducible complex characters of G. Here, we are concerned with the situation where only the cardinality of cd(G) is given. If |cd(G)| ≦ 3,, then it is known [9 ; 7] that G is solvable and the derived length dl (G) ≦ cd (G) |., If |cd(G)| = 4, then G need not be solvable (e.g., G = PSL(2, 2n))\ however [5], if G is solvable then dl(G) ≦4. 
Isaacs, I. M. Character Degrees and Derived Length of a Solvable Group. Canadian journal of mathematics, Tome 27 (1975) no. 1, pp. 146-151. doi: 10.4153/CJM-1975-018-1
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