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)

 
 

 

Groups with a character of large degree


Author: Noah Snyder
Journal: Proc. Amer. Math. Soc. 136 (2008), 1893-1903
MSC (2000): Primary 20C15
DOI: https://doi.org/10.1090/S0002-9939-08-09147-8
Published electronically: February 13, 2008
MathSciNet review: 2383494
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ G$ be a finite group of order $ n$ and $ V$ a simple $ \mathbb{C}[G]$-module of dimension $ d$. For some nonnegative number $ e$, we have $ n=d(d+e)$. If $ e$ is small, then the character of $ V$ has unusually large degree. We fix $ e$ and attempt to classify such groups. For $ e \leq 3$ we give a complete classification. For any other fixed $ e$ we show that there are only finitely many examples.


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

  • [Ber] Berkovich, Y., ``Groups with few characters of small degrees.'' Israel J. Math. 110 (1999), 325-332. MR 1750431 (2001a:20013)
  • [Bel] Belonogov, V. A., ``Recovery of an erased row or column in a table of characters of a finite group.'' Algebra Logika 41 (2002), no. 3, 259-275, 386. MR 1934536 (2003i:20011)
  • [Bu] Burtsev, A. I., ``Groups with arithmetic restrictions for conjugacy classes of elements,'' Dep. VINITI, No. 3011-82 (1982).
  • [G] Gagola, S., ``Characters vanishing on all but two conjugacy classes.'' Pacific J. Math. 109 (1983), no. 2, 363-385. MR 721927 (85e:20009)
  • [H] Huppert, B., ``Endliche Gruppen 1.'' Springer-Verlag (1967). MR 0224703 (37:302)
  • [K] Kuisch, E., ``Sylow $ p$-subgroups of solvable Camina pairs.'' J. Algebra 156 (1993), no. 2, 395-406. MR 1216476 (94e:20030)
  • [K-vdW1] Kuisch, E., and van der Waall, R., ``Homogeneous character induction.'' J. Algebra 149 (1992), no. 2, 454-471. MR 1172440 (93f:20012)
  • [K-vdW2] Kuisch, E., and van der Waall, R., ``Homogeneous character induction. II.'' J. Algebra 170 (1994), no. 2, 584-595. MR 1302857 (95i:20013)
  • [R] Robinson, D., ``A Course in the Theory of Groups.'' Springer-Verlag (1996). MR 1357169 (96f:20001)
  • [Se] Serre, J-P., ``Linear Representations of Finite Groups.'' Springer-Verlag (1977). MR 0450380 (56:8675)
  • [T] Taussky, O., ``A remark on the class field tower.'' J. London Math. Soc. 12 (1937), 82-85.
  • [Z] Zassenhaus, H., ``Über endliche Fastkörper.'' Abhandl. Hamburg 11 (1935), 187-220.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 20C15

Retrieve articles in all journals with MSC (2000): 20C15


Additional Information

Noah Snyder
Affiliation: Department of Mathematics, University of California, Berkeley, California 94720
Email: nsnyder@math.berkeley.edu

DOI: https://doi.org/10.1090/S0002-9939-08-09147-8
Received by editor(s): May 31, 2006
Published electronically: February 13, 2008
Additional Notes: This material is based upon work supported under a National Science Foundation Research Fellowship.
Communicated by: Jonathan I. Hall
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society