Bounding graph diameters of solvable groups.
For a finite group \(G\) let \(\text{cd}(G)=\{\chi(1)\mid\chi\in\text{Irr}_\mathbb{C}(G)\}\). Associated to \(\text{cd}(G)\) we define two graphs \(\Gamma(G)\) and \(\Delta(G)\). The vertices of \(\Gamma(G)\) are \(\text{cd}(G)\setminus\{1\}\) and two vertices \(\chi(1)\) and \(\psi(1)\) are joined by an edge if \(\gcd(\chi(1),\psi(1))>1\). The graph \(\Delta(G)\) has \(\{p\mid p\) a prime and \(p\mid\chi(1)\) for some \(\chi\in\text{Irr}_\mathbb{C}(G)\}\) as the set of vertices and two primes \(p,q\) are adjacent if there is \(\chi(1)\in\text{cd}(G)\) with \(pq\mid\chi(1)\). Both graphs share some properties in common, for instance the number of connected components. Thus for many problems it does not matter which graph we choose for our investigations. In 1989 Manz, Wolf and the reviewer proved that for a solvable group \(G\) the diameter \(\text{diam}(\Delta(G))\) of a connected component is always bounded by 3 and recently (2002) Lewis could show that this bound is tight. From this, easy arguments lead to \(\text{diam}(\Gamma(G))\leq 4\). However, in the paper under review the author proves the better bound \(\text{diam}(\Gamma(G))\leq 3\), in parallel to the other graph \(\Delta(G)\). In the last section examples show that there are also interesting questions for which the parallelism between the two graphs fails.
scientific article; zbMATH DE number 2136727