Groups which do not have four irreducible characters of degrees divisible by a prime \(p\). (English) Zbl 1502.20005
Summary: Given a finite group \(G\), we say that \(G\) has property \(\mathcal{P}_n\) if for every prime integer \(p\), \(G\) has at most \(n - 1\) irreducible characters whose degrees are multiples of \(p\). In this paper, we classify all finite groups that have property \(\mathcal{P}_4\). We show that the groups satisfying property \(\mathcal{P}_4\) are exactly the finite groups with at most three nonlinear irreducible characters, one solvable group of order 168, \(\mathrm{SL}_2(3)\), \(A_5\), \(S_5\), \(\mathrm{PSL}_2(7)\) and \(A_6\).
MSC:
| 20C15 | Ordinary representations and characters |
| 20D60 | Arithmetic and combinatorial problems involving abstract finite groups |
Software:
GAPReferences:
| [1] | Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; Wilson, R. A., Atlas of finite groups: maximal subgroups and ordinary characters for simple groups, (1985), Clarendon Press Oxford, England · Zbl 0568.20001 |
| [2] | Benjamin, D., Coprimeness among irreducible character degrees of finite solvable groups, Proc. Am. Math. Soc., 125, 10, 2831-2837, (1997) · Zbl 0889.20004 |
| [3] | Berkovich, Y.; Chillag, D.; Herzog, M., Finite groups in which the degrees of nonlinear irreducible characters are distinct, Proc. Am. Math. Soc., 115, 4, 955-959, (1992) · Zbl 0822.20004 |
| [4] | Berkovich, Y., Finite solvable groups in which only two nonlinear irreducible characters have equal degrees, J. Algebra, 184, 584-603, (1996) · Zbl 0861.20008 |
| [5] | Berkovich, Y. G.; Zhumd’, E. M., Characters of finite groups. part 2, Translations of Mathematical Monographs, vol. 181, (1999), AMS Providence, RI |
| [6] | Dornhoff, L., Group representation theory, part A: ordinary representation theory, (1971), Dekker New York · Zbl 0227.20002 |
| [7] | Fang, M.; Zhang, P., Finite groups with graphs containing no triangles, J. Algebra, 264, 613-619, (2003) · Zbl 1024.20021 |
| [8] | GAP - groups, algorithms, and programming, (2017), Version 4.8.7 |
| [9] | Ghaffarzadeh, M.; Ghasemi, M.; Lewis, M. L.; Tong-Viet, H. P., Nonsolvable groups with no prime dividing four character degrees, Algebr. Represent. Theory, 20, 547-567, (2017) · Zbl 1422.20003 |
| [10] | Huppert, B., Character theory of finite groups, (1998), Walter de Gruyter Berlin · Zbl 0932.20007 |
| [11] | Isaacs, I. M., Character theory of finite groups, (1976), Academic Press San Diego, California · Zbl 0337.20005 |
| [12] | Li, T.; Liu, Y.; Song, X., Finite nonsolvable groups whose character graphs have no triangle, J. Algebra, 323, 2290-2300, (2010) · Zbl 1207.20005 |
| [13] | Manz, O.; Staszewski, R.; Willems, W., On the number of components of a graph related to character degrees, Proc. Am. Math. Soc., 103, 31-37, (1988) · Zbl 0645.20005 |
| [14] | Moretó, A.; Qian, G.; Shi, W., Finite groups whose conjugacy class graphs have few vertices, Arch. Math., 85, 101-107, (2005) · Zbl 1104.20027 |
| [15] | Qian, G.; Wang, Y.; Wei, H., Finite solvable groups with at most two nonlinear irreducible characters of each degree, J. Algebra, 320, 3172-3186, (2008) · Zbl 1162.20007 |
| [16] | Qian, G.; Yang, Y., Nonsolvable groups with no prime dividing three character degrees, J. Algebra, 436, 145-160, (2015) · Zbl 1325.20005 |
| [17] | Suzuki, M., On a class of doubly transitive groups, Ann. Math., 75, 105-145, (1962) · Zbl 0106.24702 |
| [18] | Wu, Y.; Zhang, P., Finite solvable groups whose character graphs are trees, J. Algebra, 308, 536-544, (2007) · Zbl 1118.20012 |
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.
