Hostname: page-component-848d4c4894-4rdrl Total loading time: 0 Render date: 2024-07-01T08:48:47.168Z Has data issue: false hasContentIssue false
  • English
  • Français

Real class sizes and real character degrees

Published online by Cambridge University Press:  22 June 2010

ROBERT M. GURALNICK ,
GABRIEL NAVARRO  and
PHAM HUU TIEP
ROBERT M. GURALNICK
Affiliation:
Department of Mathematics, University of Southern California, Los Angeles, CA 90089-2532, U.S.A. e-mail: guralnic@math.usc.edu
GABRIEL NAVARRO
Affiliation:
Departament d'Àlgebra, Universitat de València, 46100 Burjassot, València, Spain. e-mail: gabriel@uv.es
PHAM HUU TIEP
Affiliation:
Department of Mathematics, University of Arizona, Tucson, AZ 32611-0089, U.S.A. e-mail: tiep@math.arizona.edu
Get access Rights & Permissions [Opens in a new window]

Extract

Perhaps unexpectedly, there is a rich and deep connection between field of values of characters, their degrees and the structure of a finite group. Some of the fundamental results on the degrees of characters of finite groups, as the Ito–Michler and Thompson's theorems, admit a version involving only characters with certain fixed field of values ([DNT, NS, NST2, NT1, NT3]).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 150 , Issue 1 , January 2011 , pp. 47 - 71
Copyright
Copyright © Cambridge Philosophical Society 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[CM]Chillag, D. and Mann, A.Nearly odd-order and nearly real finite groups. Comm. Algebra 26 (1998), 2041–2064.CrossRefGoogle Scholar
[Atlas]Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A. and Wilson, R. A.An ATLAS of Finite Groups (Clarendon Press, 1985).Google Scholar
[D]Deriziotis, D. I.Conjugacy Classes and Centralizers of Semisimple Elements in Finite Groups of Lie Type Vorlesungen aus dem Fachbereich Mathematik der Universität Essen, Heft 11 (1984).Google Scholar
[DNT]Dolfi, S., Navarro, G. and Tiep, P. H.Primes dividing the degrees of the real characters of finite groups. Math. Z. 259 (2008), 755–774.CrossRefGoogle Scholar
[FLT]Fleischmann, P., Lempken, W. and Tiep, P. H.Finite p′-semiregular groups. J. Algebra 188 (1997), 547–579.CrossRefGoogle Scholar
[GT1]Guralnick, R. M. and Tiep, P. H.Cross characteristic representations of even characteristic symplectic groups. Trans. Amer. Math. Soc. 356 (2004), 4969–5023.CrossRefGoogle Scholar
[GT2]Guralnick, R. M. and Tiep, P. H.The non-coprime k(GV)-problem. J. Algebra 293 (2005), 185–242.CrossRefGoogle Scholar
[I]Isaacs, I. M.Character Theory of Finite Groups (AMS-Chelsea, Providence, 2006).Google Scholar
[IN]Isaacs, I. M. and Navarro, G.Group elements and field of character of values. J. Group Theory 12 (2009), 635–650.CrossRefGoogle Scholar
[JLPW]Jansen, C., Lux, K., Parker, R. A. and Wilson, R. A.An ATLAS of Brauer Characters (Oxford University Press, 1995).Google Scholar
[KS]Kurzweil, H. and Stellmacher, B.Theorie der endlichen Gruppen (Springer, 1998).Google Scholar
[MT]Moretó, A. and Tiep, P. H.Prime divisors of character degrees. J. Group Theory 11 (2008), 341–356.CrossRefGoogle Scholar
[NS]Navarro, G. and Sanus, L.Rationality and normal 2-complements. J. Algebra 320 (2008) 2451–2454.CrossRefGoogle Scholar
[NST1]Navarro, G., Sanus, L. and Tiep, P. H.Groups with two real Brauer characters. J. Algebra 307 (2007) 891–898.Google Scholar
[NST2]Navarro, G., Sanus, L. and Tiep, P. H.Real characters and degrees. Israel J. Math. 171 (2009), 157–173.CrossRefGoogle Scholar
[NT1]Navarro, G. and Tiep, P. H.Characters of p′-degree with cyclotomic field of values. Proc. Amer Math. Soc. 134 (2006), 2833–2837.CrossRefGoogle Scholar
[NT2]Navarro, G. and Tiep, P. H.Rational irreducible characters and rational conjugacy classes in finite groups. Trans. Amer. Math. Soc. 360 (2008), 2443–2465.Google Scholar
[NT3]Navarro, G. and Tiep, P. H.Degrees of rational characters of finite groups. Adv. Math. 224 (2010), 1121–1142.Google Scholar
[S]Seress, A.Primitive groups with no regular orbits on the set of subsets. Bull. London Math. Soc. 29 (1997), 697–704.Google Scholar
[TZ1]Tiep, P. H. and Zalesskii, A. E.Minimal characters of the finite classical groups. Comm. Algebra 24 (1996), 2093–2167.CrossRefGoogle Scholar
[TZ2]Tiep, P. H. and Zalesskii, A. E.Real conjugacy classes in algebraic groups and finite groups of Lie type. J. Group Theory 8 (2005), 291–315.Google Scholar
[Zs]Zsigmondy, K.Zur Theorie der Potenzreste. Monath. Math. Phys. 3 (1892), 265–284.Google Scholar