The Taketa Problem and Character Degree Graphs with Diameter Three

Abstract

Let G be a solvable group and let Δ(G) be the character degree graph of G. The vertices of Δ(G) are the primes dividing character degrees of G and there is an edge between two primes if they divide a common character degree of G. In this paper, we show that the Taketa inequality dl(G) ≤ | cd(G)| holds when G is a solvable group whose degree graph Δ(G) has diameter 3.