A solvable group whose character degree graph has diameter \(3\) (Q2759005)

For a finite group \(G\), let \(\text{cd}(G)=\{\chi(1)\mid\chi\in\text{Irr}(G)\}\). Let \(\rho(G)\) be the set of all prime divisors of the number \(\prod_{n\in\text{cd}(G)}n\). With the set \(\rho(G)\) we associate a graph \(\Delta(G)\), the character degree graph of \(G\). The vertex set of \(\Delta(G)\) is \(\rho(G)\), and there is an edge in \(\Delta(G)\) between \(p\) and \(q\) in \(\rho(G)\) if \(pq\) divides some degree \(a\in\text{cd}(G)\). Note that \(p\in\rho(G)\) unless \(P\in\text{Syl}_p(G)\) is normal in \(G\) and Abelian. If \(G=A_5\), the graph \(\Delta(G)\) is presented by three isolated vertices.NEWLINENEWLINENEWLINELet \(G\) be a solvable group. Then the graph \(\Delta(G)\) has at most two connected components. In this note only solvable groups \(G\) with connected graph \(\Delta(G)\) are considered. Huppert asked the question: ``Is the diameter of \(\Delta(G)\) always at most \(2\)?'' The author presents a solvable group \(G\) whose degree graph has diameter \(3\). In his example, \(|\rho(G)|=6\). In an unpublished manuscript, the author proves that \(\rho(G)\geq 6\) for solvable groups \(G\) with degree graph of diameter \(3\).