Bounding Fitting heights of character degree graphs (Q5945140)

Let \(\text{cd}(G)\) be the set of (irreducible) character degrees of \(G\), and \(\rho(G)\) the set of primes dividing \(\prod_{n\in\text{cd}(G)}n\). The degree graph of \(G\), written \(\Delta(G)\), is the graph whose vertex set is \(\rho(G)\). Two vertices \(p\) and \(q\) in \(\rho(G)\) are adjacent if there is some degree \(a\in\text{cd}(G)\) where \(pq\) divides \(a\).NEWLINENEWLINENEWLINETheorems A and B: Let \(G\) be a solvable group. A. Suppose that \(\rho(G)=\pi_1\cup\pi_2\cup\{p\}\) is a disjoint union, where \(|\pi_i|\geq 1\) for \(i=1,2\). Assume that no prime in \(\pi_1\) is adjacent in \(\Delta(G)\) to any prime in \(\pi_2\). Then the Fitting height of \(G\) is at most \(4\). B. Suppose that \(\Delta(G)\) is the graph having four vertices where every vertex has degree \(2\). Then the Fitting height of \(G\) is at most \(4\).NEWLINENEWLINENEWLINELemmas 1 and 2 of this note are of independent interest.