Diameters of degree graphs of nonsolvable groups.
Let \(G\) be a finite group. The character degree graph \(\Delta(G)\) is the graph whose vertices are the primes dividing at least one irreducible complex character degree of \(G\) and whose edges are between primes whose product divides at least one such degree. This graph has been studied in quite some detail in the recent past. In the paper under review, the authors are concerned with the diameter of \(\Delta(G)\) (which is the maximum of the diameters of the connected components of \(\Delta(G)\)). They prove that it is bounded above by 2 for nonabelian finite simple groups, except for the Janko group \(J_1\), for which it is 3. They also prove that for an arbitrary finite group the diameter is bounded above by 4; they conjecture, however, that 3 is the best possible upper bound.
scientific article; zbMATH DE number 2153354
scientific article