## Rational irreducible characters and rational conjugacy classes in finite groups

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- by Gabriel Navarro and Pham Huu Tiep PDF
- Trans. Amer. Math. Soc.
**360**(2008), 2443-2465 Request permission

## Abstract:

We prove that a finite group $G$ has two rational-valued irreducible characters if and only if it has two rational conjugacy classes, and determine the structure of any such group. Along the way we also prove a conjecture of Gow stating that any finite group of even order has a non-trivial rational-valued irreducible character of odd degree.## References

- A. Borel, R. Carter, C. W. Curtis, N. Iwahori, T. A. Springer, R. Steinberg,
*Seminar on Algebraic Groups and Related Finite Groups, Lect. Notes in Math.*, Springer-Verlag, Berlin, 1970.**131** - E. G. Bryukhanova,
*Automorphism groups of $2$-automorphic $2$-groups*, Algebra i Logika**20**(1981), no. 1, 5–21, 123 (Russian). MR**635647** - Roger W. Carter,
*Finite groups of Lie type*, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1985. Conjugacy classes and complex characters; A Wiley-Interscience Publication. MR**794307** - J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson,
*$\Bbb {ATLAS}$ 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** - P. Deligne and G. Lusztig,
*Representations of reductive groups over finite fields*, Ann. of Math. (2)**103**(1976), no. 1, 103–161. MR**393266**, DOI 10.2307/1971021 - François Digne and Jean Michel,
*Representations of finite groups of Lie type*, London Mathematical Society Student Texts, vol. 21, Cambridge University Press, Cambridge, 1991. MR**1118841**, DOI 10.1017/CBO9781139172417 - Larry Dornhoff,
*Group representation theory. Part A: Ordinary representation theory*, Pure and Applied Mathematics, vol. 7, Marcel Dekker, Inc., New York, 1971. MR**0347959** - Fletcher Gross,
*$2$-automorphic $2$-groups*, J. Algebra**40**(1976), no. 2, 348–353. MR**409642**, DOI 10.1016/0021-8693(76)90199-X - Robert M. Guralnick and Jan Saxl,
*Generation of finite almost simple groups by conjugates*, J. Algebra**268**(2003), no. 2, 519–571. MR**2009321**, DOI 10.1016/S0021-8693(03)00182-0 - Robert M. Guralnick and Pham Huu Tiep,
*Cross characteristic representations of even characteristic symplectic groups*, Trans. Amer. Math. Soc.**356**(2004), no. 12, 4969–5023. MR**2084408**, DOI 10.1090/S0002-9947-04-03477-4 - Robert M. Guralnick and Pham Huu Tiep,
*The non-coprime $k(GV)$ problem*, J. Algebra**293**(2005), no. 1, 185–242. MR**2173972**, DOI 10.1016/j.jalgebra.2005.02.001 - Bertram Huppert and Norman Blackburn,
*Finite groups. II*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 242, Springer-Verlag, Berlin-New York, 1982. AMD, 44. MR**650245** - I. M. Isaacs,
*Characters of $\pi$-separable groups*, J. Algebra**86**(1984), no. 1, 98–128. MR**727371**, DOI 10.1016/0021-8693(84)90058-9 - I. Martin Isaacs,
*Character theory of finite groups*, Dover Publications, Inc., New York, 1994. Corrected reprint of the 1976 original [Academic Press, New York; MR0460423 (57 #417)]. MR**1280461** - Shiro Iwasaki,
*On finite groups with exactly two real conjugate classes*, Arch. Math. (Basel)**33**(1979/80), no. 6, 512–517. MR**570486**, DOI 10.1007/BF01222794 - Christoph Jansen, Klaus Lux, Richard Parker, and Robert Wilson,
*An atlas of Brauer characters*, London Mathematical Society Monographs. New Series, vol. 11, The Clarendon Press, Oxford University Press, New York, 1995. Appendix 2 by T. Breuer and S. Norton; Oxford Science Publications. MR**1367961** - George Lusztig,
*Characters of reductive groups over a finite field*, Annals of Mathematics Studies, vol. 107, Princeton University Press, Princeton, NJ, 1984. MR**742472**, DOI 10.1515/9781400881772 - G. Lusztig,
*On the representations of reductive groups with disconnected centre*, Astérisque**168**(1988), 10, 157–166. Orbites unipotentes et représentations, I. MR**1021495** - Martin W. Liebeck,
*The affine permutation groups of rank three*, Proc. London Math. Soc. (3)**54**(1987), no. 3, 477–516. MR**879395**, DOI 10.1112/plms/s3-54.3.477 - Frank Lübeck,
*Smallest degrees of representations of exceptional groups of Lie type*, Comm. Algebra**29**(2001), no. 5, 2147–2169. MR**1837968**, DOI 10.1081/AGB-100002175 - I. M. Richards,
*Characters of groups with quotients of odd order*, J. Algebra**96**(1985), no. 1, 45–47. MR**808839**, DOI 10.1016/0021-8693(85)90037-7 - William A. Simpson and J. Sutherland Frame,
*The character tables for $\textrm {SL}(3,\,q)$, $\textrm {SU}(3,\,q^{2})$, $\textrm {PSL}(3,\,q)$, $\textrm {PSU}(3,\,q^{2})$*, Canadian J. Math.**25**(1973), 486–494. MR**335618**, DOI 10.4153/CJM-1973-049-7 - Peter Sin and Pham Huu Tiep,
*Rank 3 permutation modules of the finite classical groups*, J. Algebra**291**(2005), no. 2, 551–606. MR**2163483**, DOI 10.1016/j.jalgebra.2005.02.005 - Bettina Wilkens,
*A note on $2$-automorphic $2$-groups*, J. Algebra**184**(1996), no. 1, 199–206. MR**1402576**, DOI 10.1006/jabr.1996.0255 - Thomas R. Wolf,
*Character correspondences in solvable groups*, Illinois J. Math.**22**(1978), no. 2, 327–340. MR**498821**

## Additional Information

**Gabriel Navarro**- Affiliation: Facultat de Matemàtiques, Universitat de València, Burjassot, València 46100, Spain
- MR Author ID: 129760
- Email: gabriel@uv.es
**Pham Huu Tiep**- Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611
- MR Author ID: 230310
- Email: tiep@math.ufl.edu
- Received by editor(s): February 6, 2006
- Published electronically: November 27, 2007
- Additional Notes: The first author was partially supported by the Ministerio de Educación y Ciencia proyecto MTM2004-06067-C02-01.

Part of this work was done while the first author visited the University of Florida in Gainesville, and he would like to thank the Mathematics Department for its hospitality. Special thanks are due to A. Turull. This paper benefited from conversations with M. Isaacs, A. Moretó, A. Turull and B. Wilkens. The authors are grateful to the referee for pointing out some inaccuracies in an earlier version of the paper as well as for helpful comments that greatly improved the exposition of the paper.

The second author gratefully acknowledges the support of the NSA and the NSF - © Copyright 2007 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**360**(2008), 2443-2465 - MSC (2000): Primary 20C15, 20C33, 20E45
- DOI: https://doi.org/10.1090/S0002-9947-07-04375-9
- MathSciNet review: 2373321

Dedicated: Dedicated to Professor Michel Broué on the occasion of his sixtieth birthday