Let \(h(G)\) be the Fitting height (or nilpotent length) of a solvable group \(G\). As S. Garrison showed, if \(G\) is solvable, then \(h(G)\leq|\text{cd}(G)|\), where \(\text{cd}(G)=\{\chi(1)\mid\chi\in\text{Irr}(G)\}\) [see Corollary 12.21 in I. M. Isaacs, Character theory of finite groups, Academic Press, New York (1976; Zbl 0337.20005)]. However, in the author’s thesis it is proved that if \(G\) is solvable and \(|\text{cd}(G)|\geq 4\), then \(h(G)\leq|\text{cd}(G)|-1\), and this result is best possible. There exists a group \(G=H\cdot V\), where \(V\) is elementary Abelian of order \(q^2\), \(q\equiv 1\pmod{16}\), and \(H\) is a subgroup of order \(8\) in \(\text{SL}(2,q)\) such that \(\text{cd}(G)=\{1,2,3,4,48\}\) and \(h(G)=4\). As Theorem A asserts, if \(G\) is \(2\)-closed and \(|\text{cd}(G)|=5\), then \(h(G)\leq 3\). It follows from the above theorem that, if \(G\) is \(2\)-closed and \(|\text{cd}(G)|\geq 5\), then \(h(G)\leq|\text{cd}(G)|-2\).
Theorem 1.3. Suppose that a Sylow \(2\)-subgroup of \(G\) is Abelian and normal in \(G\). Suppose that \(\text{cd}(G)=\{1=d_1,d_2,\dots,d_m\}\) with \(d_1<\cdots<d_m\). Then the \(i\)-th derived subgroup of \(G\) is contained in \(\ker(\chi)\) for all \(\chi\in\text{Irr}(G)\) satisfying \(\chi(1)\leq d_i\). In particular, the derived length of \(G\) is at most \(m\).
T. R. Berger [J. Algebra 39, 199-207 (1976; Zbl 0362.20012)] has proved the above theorem for groups of odd order. As the author notes, Berger’s original argument still works under the hypothesis of Theorem 1.3.