Previous |  Up |  Next

Article

Full entry | Fulltext not available (moving wall 24 months)      Feedback
Keywords:
non-solvable group; solvable group; character degree
Summary:
Let $m>1$ be a fixed positive integer. In this paper, we consider finite groups each of whose nonlinear character degrees has exactly $m$ prime divisors. We show that such groups are solvable whenever $m>2$. Moreover, we prove that if $G$ is a non-solvable group with this property, then $m=2$ and $G$ is an extension of ${\rm A}_7$ or ${\rm S}_7$ by a solvable group.
References:
[1] Benjamin, D.: Coprimeness among irreducible character degrees of finite solvable groups. Proc. Am. Math. Soc. 125 (1997), 2831-2837. DOI 10.1090/S0002-9939-97-04269-X | MR 1443370 | Zbl 0889.20004
[2] Bianchi, M., Chillag, D., Lewis, M. L., Pacifici, E.: Character degree graphs that are complete graphs. Proc. Am. Math. Soc. 135 (2007), 671-676. DOI 10.1090/S0002-9939-06-08651-5 | MR 2262862 | Zbl 1112.20006
[3] 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. Clarendon Press, Oxford (1985). MR 0827219 | Zbl 0568.20001
[4] Group, GAP: GAP -- Groups, Algorithms, and Programming. A System for Computational Discrete Algebra. Version 4.7.4. Available at http://www.gap-system.org (2014).
[5] Isaacs, I. M.: Character Theory of Finite Groups. Pure and Applied Mathematics 69. Academic Press, New York (1976). DOI 10.1090/chel/359 | MR 0460423 | Zbl 0337.20005
[6] Isaacs, I. M., Passman, D. S.: A characterization of groups in terms of the degrees of their characters II. Pac. J. Math. 24 (1968), 467-510. DOI 10.2140/pjm.1968.24.467 | MR 0241524 | Zbl 0155.05502
[7] James, G., Kerber, A.: The Representation Theory of the Symmetric Group. Encyclopedia of Mathematics and Its Applications 16. Addison-Wesley, Reading (1981). DOI 10.1017/CBO9781107340732 | MR 644144 | Zbl 0491.20010
[8] Manz, O.: Endliche auflösbare Gruppen, deren sämtliche Charaktergrade Primzahlpotenzen sind. J. Algebra 94 (1985), 211-255 German. DOI 10.1016/0021-8693(85)90210-8 | MR 789547 | Zbl 0596.20007
[9] Manz, O.: Endliche nicht-auflösbare Gruppen, deren sämtliche Charaktergrade Primzahlpotenzen sind. J. Algebra 96 (1985), 114-119 German. DOI 10.1016/0021-8693(85)90042-0 | MR 808844 | Zbl 0567.20004
[10] Schmid, P.: Extending the Steinberg representation. J. Algebra 150 (1992), 254-256. DOI 10.1016/S0021-8693(05)80060-2 | MR 1174899 | Zbl 0794.20022
Partner of
EuDML logo