Finite nonsolvable groups whose character graphs have no triangles. (English) Zbl 1207.20005
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 O. Manz, R. Staszewski and 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 Y.-T. Wu and 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.
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 O. Manz, R. Staszewski and 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 Y.-T. Wu and 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.
Reviewer: R. W. van der Waall (Huizen)
MSC:
| 20C15 | Ordinary representations and characters |
| 05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |
| 20D60 | Arithmetic and combinatorial problems involving abstract finite groups |
| 20D05 | Finite simple groups and their classification |
Keywords:
character graphs; degrees of irreducible characters; finite groups; irreducible complex characters; connected components; alternating group \(A_5\)Software:
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