Gluck’s Conjecture

In our first major application of large orbit theorems, we discuss Gluck’s conjecture for solvable groups. If b(G) is the largest character degree of the solvable group G, then it follows relatively easily from the results in Chap. 9 that \(|G : \textbf{F}(G)| \le b(G)^3\) . We then discuss a recent improvement, using more large orbit theory, that \(|G : \textbf{F}(G)| \le b(G)^{2.6}\) .