Fitting heights and the character degree graph
Let \(G\) be a finite solvable group and \(\rho(G)=\bigcup_{\chi\in\text{Irr}(G)}\pi(\chi(1))\), where \(\pi(n)\) is the set of prime divisors of \(n\in\mathbb{N}\); let \(\text{cd}(G)=\{\chi(1)\mid\chi\in\text{Irr}(G)\}\). We can attach a graph \(\Gamma(G)\) to \(\text{cd}(G)\) called the degree graph of \(G\). The vertex set of this graph is \(\rho(G)\), and we place an edge between \(p\) and \(q\), if \(pq\) divides some degree \(a\in\text{cd}(G)\). A degree graph \(\Gamma\) has bounded Fitting height if there is a bound on the Fitting heights of the solvable groups whose degree graph is \(\Gamma\). Main theorem: Let \(\Gamma\) be a degree graph with \(n\) vertices. Then \(\Gamma\) has bounded Fitting height if and only if \(\Gamma\) has at most one vertex of degree \(n-1\). Theorems 3 and 4. Let \(G\) be solvable group with degree graph \(\Gamma=\Gamma(G)\), \(\Gamma\) has \(n\) vertices. (a) If \(\Gamma\) has at most one vertex of degree \(n-1\), then the Fitting height \(h(G)\leq 4(n-1)+2\). (b) Assume that \(\Gamma\) has distinct vertices \(u\) and \(v\) both having degree \(n-1\). Then \(\Gamma\) has unbounded Fitting height.
scientific article; zbMATH DE number 1556584