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 is a finite group such that , then has an element , such that , where is the maximal number of zeros in a row of the character table of . 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 be a finite group such that ; then has an irreducible character such that , where is the maximal number of zeros in a column of the character table of .

**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**200356**, 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**152570****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.

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