Finite nonsolvable groups whose character graphs have no triangles.
It is shown that the alternating group \(A_5\) is the only nonsolvable group whose character graph has no triangles. Here the character graph of a finite group \(G\) is defined as follows. The vertices of that graph are the nonlinear complex irreducible characters of \(G\), there is an edge between two vertices \(\chi\) and \(\psi\) if and only if \(\chi(1)\) and \(\psi(1)\) admit a common prime divisor. The authors mention that, due to \textit{O. Manz}, \textit{R. Staszewski} and \textit{W. Willems} [Proc. Am. Math. Soc. 103, No. 1, 31-37 (1988; Zbl 0645.20005)], the number of components of the character graph of any group is at most 3 and that it is at most 2 for any solvable group. Recently, due to \textit{Y.-T. Wu} and \textit{P. Zhang} [J. Algebra 308, No. 2, 536-544 (2007; Zbl 1118.20012)], a characterization of those finite solvable groups has been obtained whose character graphs have no triangles. Namely this happens precisely when the solvable group \(G\) is not isomorphic to \(S_4\) or if \(\#(\text{Irr}(G)\setminus\text{LIrr}(G))\leq 2\). Thus, by means of the paper under review and by the papers mentioned before, the classification of those finite groups whose character graphs have no triangles, is complete.
scientific article; zbMATH DE number 5696950
scientific article