Fitting Height and Character Degree Graphs

Abstract

Given a group G, (G) is the graph whose vertices are the primes that divide the degree of some irreducible character and two vertices p and q are joined by an edge if pq divides the degree of some irreducible character of G. By a definition of Lewis, a graph has bounded Fitting height if the Fitting height of any solvable group G with (G) = is bounded (in terms of ). In this note, we prove that there exists a universal constant C such that if has bounded Fitting height and (G) = then h(G) ≤ C. This solves a problem raised by Lewis.