Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On zeros of characters of finite groups


Author: David Chillag
Journal: Proc. Amer. Math. Soc. 127 (1999), 977-983
MSC (1991): Primary 20Cxx
DOI: https://doi.org/10.1090/S0002-9939-99-04790-5
MathSciNet review: 1487363
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We present several results connecting the number of conjugacy classes of a finite group on which an irreducible character vanishes, and the size of some centralizer of an element. For example, we show that if $G$ is a finite group such that $G\ne G'\ne G''$, then $G$ has an element $x$, such that $|C_G(x)|\le 2m$, where $m$ is the maximal number of zeros in a row of the character table of $G$. Dual results connecting the number of irreducible characters which are zero on a fixed conjugacy class, and the degree of some irreducible character, are included too. For example, the dual of the above result is the following: Let $G$ be a finite group such that $1\ne Z(G)\ne Z_2(G)$; then $G$ has an irreducible character $\chi$ such that $\frac{|G|}{\chi^2(1)}\le 2m$, where $m$ is the maximal number of zeros in a column of the character table of $G$.


References [Enhancements On Off] (What's this?)

  • 1. David Chillag and Marcel Herzog, On the length of the conjugacy classes of finite groups, J. Algebra 131 (1990), no. 1, 110–125. MR 1055001, https://doi.org/10.1016/0021-8693(90)90168-N
  • 2. J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, Atlas of finite groups, Oxford University Press, Eynsham, 1985. Maximal subgroups and ordinary characters for simple groups; With computational assistance from J. G. Thackray. MR 827219
  • 3. P. X. Gallagher, Zeros of characters of finite groups, J. Algebra 4 (1966), 42–45. MR 0200356, https://doi.org/10.1016/0021-8693(66)90048-2
  • 4. Stephen M. Gagola Jr., Characters vanishing on all but two conjugacy classes, Pacific J. Math. 109 (1983), no. 2, 363–385. MR 721927
  • 5. D. M. Goldsmith, Elements of order two in finite groups, Delta 4 (1974), 45-59.
  • 6. B. Huppert, Endliche Gruppen. I, Die Grundlehren der Mathematischen Wissenschaften, Band 134, Springer-Verlag, Berlin-New York, 1967 (German). MR 0224703
  • 7. I. Martin Isaacs, Character theory of finite groups, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. Pure and Applied Mathematics, No. 69. MR 0460423
  • 8. Encyclopedic dictionary of mathematics. Vol. I–IV, 2nd ed., MIT Press, Cambridge, MA, 1987. Translated from the Japanese; Edited by Kiyosi Itô. MR 901762
  • 9. Michio Suzuki, Two characteristic properties of (𝑍𝑇)-groups, Osaka Math. J. 15 (1963), 143–150. MR 0152570
  • 10. W. Willems, Blocks of defect zero and degree problems, Proc. of Symposia in Pure Math. 47 (1987), 481-484. CMP 20:10
  • 11. È. M. Zhmud′, Estimates for the number of zeros of irreducible characters of a finite group, Publ. Math. Debrecen 33 (1986), no. 1-2, 125–146 (Russian). MR 854623
  • 12. È. M. Zhmud′, On Gallagher’s theorems on zeros of group characters, Publ. Math. Debrecen 37 (1990), no. 3-4, 345–353 (Russian). MR 1082313
  • 13. E. M. Zhmud, On finite groups having an irreducible complex character with one class of zeros, Soviet Math. Dokl. 20 (1979), 795-797.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 20Cxx

Retrieve articles in all journals with MSC (1991): 20Cxx


Additional Information

David Chillag
Affiliation: Department of Mathematics, Technion, Israel Institute of Technology, Haifa 32000, Israel
Email: chillag@techunix.technion.ac.il

DOI: https://doi.org/10.1090/S0002-9939-99-04790-5
Received by editor(s): August 1, 1997
Dedicated: Dedicated to Avinoam Mann on the occasion of his 60th birthday
Communicated by: Ronald M. Solomon
Article copyright: © Copyright 1999 American Mathematical Society