Square character degree graphs yield direct products.
For any finite group \(G\), the character degree graph \(\Delta(G)\) is the graph whose vertices are the primes dividing some (irreducible complex) character degree of \(G\), and two such primes \(p,q\) are connected by an edge if \(pq\) divides some character degree. It is easy to find solvable groups whose character degree graph is a square. However, all known examples are direct products.NEWLINENEWLINE The main result of the paper under review is that indeed no other examples exist. More precisely, for a finite group \(H\) we denote by \(\rho(H)\) the set of all primes dividing some character degree of \(H\). If \(G\) is a finite solvable group for which \(\Delta(G)\) is a square, then there exist subgroups \(A\) and \(B\) of \(G\) such that \(G=A\times B\) and \(|\rho(A)|=|\rho(B)|=2\). The proof is, not unexpectedly, long and technical. The authors believe that the result remains true without the solvability hypothesis.
scientific article; zbMATH DE number 6037573