Isaacs and Seitz conjectured that the derived length of a finite solvable group $G$ is bounded by the cardinality of the set of all irreducible character degrees of $G$. We prove that the conjecture holds for $G$ if the degrees of nonlinear monolithic characters of $G$ having the same kernels are distinct. Also, we show that the conjecture is true when $G$ has at most three nonlinear monolithic characters. We give some sufficient conditions for the inequality related to monolithic characters or real-valued irreducible characters of $G$ when the commutator subgroup of $G$ is supersolvable.

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关于导出长度和字符度数的注释

Isaacs 和 Seitz 推测有限可解群的导出长度$G$是由所有不可约特征度的集合的基数限制的$G$. 我们证明该猜想成立$G$如果非线性单片特征的程度$G$具有相同内核是不同的。此外，我们证明了这个猜想是正确的，当$G$最多具有三个非线性整体特征。我们给出了与整体特征或实值不可约特征有关的不等式的一些充分条件$G$当换向子群$G$是超可解的。