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

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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.**D. Chillag and M. Herzog,*On the length of the conjugacy classes of finite groups*, J. Algebra**131**(1990), 100-125. MR**91c:20038****2.**J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson,*Atlas of Finite Groups*, Clarendon Press, 1985. MR**88g:20025****3.**P. X. Gallagher,*Zeros of characters of finite groups*, J. Algebra**4**(1966), 42-45. MR**34:252****4.**S. M. Gagola,*Characters vanishing on all but two conjugacy classes*, Pacific J. Math.**109**(1983), 363-385. MR**85e:20009****5.**D. M. Goldsmith,*Elements of order two in finite groups*, Delta**4**(1974), 45-59.**6.**B. Huppert,*Endliche Gruppen*I, Springer-Verlag, 1967. MR**37:302****7.**I. M. Isaacs,*Character Theory of Finite groups*, Academic Press, 1976. MR**57:417****8.**K. Ito (editor),*Encyclopedic Dictionary of Mathematics*, Vol. IV, Appendices and Indexes, second edition, MIT Press. MR**89b:00033****9.**M. Suzuki,*Two characteristic properties of (ZT)-groups*, Osaka Math. J.**15**(1963), 143-150. MR**27:2548****10.**W. Willems,*Blocks of defect zero and degree problems*, Proc. of Symposia in Pure Math.**47**(1987), 481-484. CMP**20:10****11.**E. M. Zhmud,*On estimates of the number of zeros of irreducible characters of a finite group*, (Russian), Publ. Math. Debrecen**33**(1986), 125-146. MR**88a:20019****12.**E. M. Zhmud,*On Gallagher's theorems on zeros of groups characters*(Russian), Publ. Math. Debrecen**37**(1990), 245-353. MR**91k:20016****13.**E. M. Zhmud,*On finite groups having an irreducible complex character with one class of zeros*, Soviet Math. Dokl.**20**(1979), 795-797.

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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