Finite solvable groups whose character graphs are trees. (English) Zbl 1118.20012
For a finite group \(G\), the character graph \(\Gamma(G)\) has the nonlinear irreducible complex characters as its vertices, and two of them are joined by an edge whenever their degrees have a common prime divisor. – This graph has been studied quite extensively in recent years.
In the paper under review it is shown that for solvable groups, \(\Gamma(G)\) has no triangles if and only if \(\Gamma(G)\) has only two vertices, or \(G\) is isomorphic to the symmetric group \(S_4\). The condition of having no triangles is also equivalent to \(\Gamma(G)\) being a tree.
In the paper under review it is shown that for solvable groups, \(\Gamma(G)\) has no triangles if and only if \(\Gamma(G)\) has only two vertices, or \(G\) is isomorphic to the symmetric group \(S_4\). The condition of having no triangles is also equivalent to \(\Gamma(G)\) being a tree.
Reviewer: Thomas Michael Keller (San Marcos)
MSC:
| 20C15 | Ordinary representations and characters |
| 20D60 | Arithmetic and combinatorial problems involving abstract finite groups |
| 05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |
| 20D10 | Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks |
Keywords:
finite groups; character degree graphs; trees; irreducible complex characters; symmetric group \(S_4\)References:
| [1] | Berkovich, Y.; Chillag, D.; Herzog, M., Finite groups in which the degrees of the nonlinear irreducible characters are distinct, Proc. Amer. Math. Soc., 115, 4, 955-959 (1992), MR1088438 (92j:20006) · Zbl 0822.20004 |
| [2] | Bertram, E. A.; Herzog, M.; Mann, A., On a graph related to conjugacy classes of groups, Bull. London Math. Soc., 22, 569-575 (1990), MR1099007 (92f:20024) · Zbl 0743.20017 |
| [3] | Chillag, D.; Herzog, M.; Mann, A., On the diameter of a graph related to conjugacy classes of groups, Bull. London Math. Soc., 25, 255-262 (1993), MR1209249 (94a:20038) · Zbl 0810.20018 |
| [4] | Curtis, C.; Reiner, I., Methods of Representation Theory, vol. I (1981), Wiley, MR0632548 (82i:20001) |
| [5] | Fang, M.; Zhang, P., Finite groups with graphs containing no triangles, J. Algebra, 264, 613-619 (2003), MR1981424 (2004f:20051) · Zbl 1024.20021 |
| [6] | Isaacs, I. M., Character Theory of Finite Groups (1976), Academic Press, MR0460423 (57#417) · Zbl 0337.20005 |
| [7] | Lewis, M. L., Derived lengths and character degrees, Proc. Amer. Math. Soc., 126, 7, 1915-1921 (1998), MR1452810 (98h:20011) · Zbl 0915.20002 |
| [8] | M.L. Lewis, An overview of graphs associated with character degrees and conjugacy class sizes in finite groups, Rocky Mountain J. Math., in press; M.L. Lewis, An overview of graphs associated with character degrees and conjugacy class sizes in finite groups, Rocky Mountain J. Math., in press · Zbl 1166.20006 |
| [9] | Lu, Z. Q.; Zhang, J. P., On the diameter of a graph related to \(p\)-regular conjugacy classes of finite groups, J. Algebra, 231, 2, 705-712 (2000), MR1778167 (2001h:05051) · Zbl 0964.20009 |
| [10] | Manz, O.; Wolf, T., Representations of solvable groups, London Math. Soc. Lecture Note Ser., vol. 185 (1993), Cambridge Univ. Press: Cambridge Univ. Press Cambridge, MR1261638 (95c:20013) · Zbl 0928.20008 |
| [11] | Moretó, A.; Qian, G. H.; Shi, W. J., Finite groups whose conjugacy class graphs have few vertices, Arch. Math., 85, 2, 101-107 (2005), MR2161799 (Review) · Zbl 1104.20027 |
| [12] | Manz, O.; Staszewski, R.; Willems, W., On the number of components of a graph related to character degrees, Proc. Amer. Math. Soc., 103, 1, 31-37 (1988), MR0938639 (89b:20030) · Zbl 0645.20005 |
| [13] | Manz, O.; Willems, W.; Wolf, T. T., The diameter of the character degree graph, J. Reine Angew. Math., 402, 181-198 (1989), MR1022799 (90i:20007) · Zbl 0678.20002 |
| [14] | Pálfy, P. P., Groups with two nonlinear irreducible representations, Ann. Uni. Sci. Budapest. Eötvös Sect. Math., 24, 181-192 (1981), MR0878225 (88a:20018) · Zbl 0485.20008 |
| [15] | Seitz, G., Finite groups having only one irreducible representation of degree greater than one, Proc. Amer. Math. Soc., 19, 459-461 (1968), MR0222160 (36#5212) · Zbl 0244.20010 |
| [16] | Zhang, G. X., Groups with two non-linear irreducible characters, Chinese J. Contemp. Math., 17, 2, 175-184 (1996), MR1432498 (98b:20012) |
| [17] | Zhang, J. P., A note on character degrees of finite solvable groups, Comm. Algebra, 28, 9, 4249-4258 (2000), MR1772504 (2001f:20020) · Zbl 0980.20005 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
