## On a theorem of Feit and Tits

HTML articles powered by AMS MathViewer

- by Peter B. Kleidman and Martin W. Liebeck PDF
- Proc. Amer. Math. Soc.
**107**(1989), 315-322 Request permission

## Abstract:

Feit and Tits [3] lay the groundwork for determining the smallest degree of a projective representation of a finite extension of a finite simple group $G$. Provided $G$ is not of Lie type in characteristic 2, they determine precisely when this degree is smaller than the degree of a projective representation of $G$ itself. We complete this project by extending their results to the groups of Lie type in characteristic 2.## References

- 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** - Hikoe Enomoto and Hiromichi Yamada,
*The characters of $G_2(2^n)$*, Japan. J. Math. (N.S.)**12**(1986), no. 2, 325–377. MR**914301**, DOI 10.4099/math1924.12.325 - Walter Feit and Jacques Tits,
*Projective representations of minimum degree of group extensions*, Canadian J. Math.**30**(1978), no. 5, 1092–1102. MR**498824**, DOI 10.4153/CJM-1978-092-5 - J. A. Green,
*The characters of the finite general linear groups*, Trans. Amer. Math. Soc.**80**(1955), 402–447. MR**72878**, DOI 10.1090/S0002-9947-1955-0072878-2 - Robert L. Griess Jr.,
*Automorphisms of extra special groups and nonvanishing degree $2$ cohomology*, Pacific J. Math.**48**(1973), 403–422. MR**476878** - Gerhard Hiss,
*On the decomposition numbers of $G_2(q)$*, J. Algebra**120**(1989), no. 2, 339–360. MR**989902**, DOI 10.1016/0021-8693(89)90201-9
G. Hiss and J. Shamash, - I. Martin Isaacs,
*Character theory of finite groups*, Pure and Applied Mathematics, No. 69, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. MR**0460423**
G. D. James, - Vicente Landazuri and Gary M. Seitz,
*On the minimal degrees of projective representations of the finite Chevalley groups*, J. Algebra**32**(1974), 418–443. MR**360852**, DOI 10.1016/0021-8693(74)90150-1 - Martin W. Liebeck,
*On the orders of maximal subgroups of the finite classical groups*, Proc. London Math. Soc. (3)**50**(1985), no. 3, 426–446. MR**779398**, DOI 10.1112/plms/s3-50.3.426 - 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 - Gary M. Seitz,
*Some representations of classical groups*, J. London Math. Soc. (2)**10**(1975), 115–120. MR**369556**, DOI 10.1112/jlms/s2-10.1.115 - Josephine Shamash,
*Blocks and Brauer trees for groups of type $G_2(q)$*, The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986) Proc. Sympos. Pure Math., vol. 47, Amer. Math. Soc., Providence, RI, 1987, pp. 283–295. MR**933418** - Robert Steinberg,
*Lectures on Chevalley groups*, Yale University, New Haven, Conn., 1968. Notes prepared by John Faulkner and Robert Wilson. MR**0466335** - K. Zsigmondy,
*Zur Theorie der Potenzreste*, Monatsh. Math. Phys.**3**(1892), no. 1, 265–284 (German). MR**1546236**, DOI 10.1007/BF01692444

*The*$3$

*-modular representations of*${G_2}({2^n})$ (to appear).

*The decomposition matrices for*${\text {G}}{{\text {L}}_n}(q),n \leq 10$, Imperial College, London, (preprint).

## Additional Information

- © Copyright 1989 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**107**(1989), 315-322 - MSC: Primary 20C25
- DOI: https://doi.org/10.1090/S0002-9939-1989-0961412-6
- MathSciNet review: 961412