Character degree graphs with no complete vertices.
Let \(G\) be a finite group. The character degree graph \(\Gamma(G)\) of \(G\) is the graph with vertex set \(\rho(G)=\{p\mid p\text{ a prime, }p\mid\chi(1)\text{ for some }\chi\in\text{Irr}(G)\}\) and edges between two elements \(p\) and \(q\) of \(\rho(G)\) if and only if \(pq\) divides some irreducible character degree of \(G\).NEWLINENEWLINE The purpose of the article under review is to establish whether certain graphs can be character degree graphs of solvable groups and to obtain a characterization of the groups that admit them. The main result is the following. Let \(G\) be solvable with Abelian Fitting group \(F(G)\). Suppose that the degree in \(\Gamma(G)\) of every \(\upsilon\in\rho(G)\) is at most \(|\rho(G)-2|\). Then there is an integer \(n\) such that \(F(G)=M_1\times\cdots\times M_n\times Z\), where \(M_1,\dots,M_n\) are minimal normal subgroups of \(G\) and \(Z=Z(G)\); moreover \(G=D_1\times\cdots\times D_n\), where \(M_1\leq D_1,\dots,M_n\leq D_n\), \((|D_i/M_i|,|D_j/M_j|)=1\) for all \(i\neq j\), and \(\Gamma(D_1),\dots,\Gamma(D_n)\) are disconnected graphs.NEWLINENEWLINE The author draws some consequences of this result. For example, if \(G\) is a solvable group with Abelian Fitting subgroup and connected character degree graph, then the diameter of \(\Gamma(G)\) is at most 2. (The general best possible bound for the diameter in the case of solvable groups is known to be 3.)
scientific article; zbMATH DE number 6070505
scientific article