Irreducible character degrees and derived length

K. Taketa has proved that all M-groups are solvable in [35]. The proof of this fact shows that if G is an M-group, then the derived length of G is at most the cardinality of the set of all irreducible character degrees of G. Then, it has been conjectured by I. M. Isaacs and G. Seitz that the inequality dl(G) ≤ cd(G) holds for arbitrary finite solvable group G, where dl(G) is the derived length of G and cd(G) is the set of all irreducible complex character degrees of G. Although this conjecture is still open, in the literature there are several results which prove that some classes of solvable groups satisfy this inequality. In this thesis, we prove that the Taketa inequality holds for a finite solvable group G if monolithic or real irreducible characters of G have some special properties. We also present relations between the structure of a finite solvable group G and the kernels of irreducible constituents of the character where is an irreducible character of G and thus we obtain some results for the Taketa inequality.