Bounding graph diameters of nonsolvable groups.
For a finite group \(G\) the set of degrees of all the irreducible characters of \(G\) is denoted by \(\text{cd}(G)\). The common-divisor graph \(\Gamma(G)\) associated with \(G\) is a graph with vertices \(\chi(1)\in\text{cd}(G)\) different from 1 and two vertices \(a\) and \(b\) are adjacent if they have a non-trivial common divisor. The author proved [in J. Algebra 280, No. 2, 415-425 (2004; Zbl 1077.20010)] that if \(G\) is a solvable group, then the diameter of \(\Gamma(G)\) is at most \(3\). In the paper under review the author proves the same result for a non-solvable group, extending the same result for every finite group.
scientific article; zbMATH DE number 2119564
scientific article