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)

 

 

Schur indices of perfect groups


Author: Alexandre Turull
Journal: Proc. Amer. Math. Soc. 130 (2002), 367-370
MSC (2000): Primary 20C15
Published electronically: June 8, 2001
MathSciNet review: 1862114
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract:

It has been noticed by many authors that the Schur indices of the irreducible characters of many quasi-simple finite groups are at most $2$. A conjecture has emerged that the Schur indices of all irreducible characters of all quasi-simple finite groups are at most $2$. We prove that this conjecture cannot be extended to the set of all finite perfect groups. Indeed, we prove that, given any positive integer $n$, there exist irreducible characters of finite perfect groups of chief length $2$ which have Schur index $n$.


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

  • [1] R. Brauer, Untersuchungen über die arithmetischen Eigenschaften von Gruppen Substitutionen II, Math. Z. 31 (1930), 733-747.
  • [2] Walter Feit, Characters of finite groups, W. A. Benjamin, Inc., New York-Amsterdam, 1967. MR 0219636
  • [3] Walter Feit, The computations of some Schur indices, Israel J. Math. 46 (1983), no. 4, 274–300. MR 730344, 10.1007/BF02762888
  • [4] Walter Feit, Schur indices of characters of groups related to finite sporadic simple groups, Israel J. Math. 93 (1996), 229–251. MR 1380645, 10.1007/BF02761105
  • [5] Walter Feit and Gregg J. Zuckerman, Reality properties of conjugacy classes in spin groups and symplectic groups, Algebraists’ homage: papers in ring theory and related topics (New Haven, Conn., 1981) Contemp. Math., vol. 13, Amer. Math. Soc., Providence, R.I., 1982, pp. 239–253. MR 685957
  • [6] R. Gow, On the Schur indices of characters of finite classical groups, J. London Math. Soc. (2) 24 (1981), no. 1, 135–147. MR 623680, 10.1112/jlms/s2-24.1.135
  • [7] B. Huppert, Endliche Gruppen. I, Die Grundlehren der Mathematischen Wissenschaften, Band 134, Springer-Verlag, Berlin-New York, 1967 (German). MR 0224703
  • [8] Zyozyu Ohmori, On the Schur indices of certain irreducible characters of simple algebraic groups over finite fields, Proc. Japan Acad. Ser. A Math. Sci. 64 (1988), no. 7, 253–255. MR 974086
  • [9] Z. Ohmori, On the upper bounds of the Schur indices of simple finite groups of Lie type, Proc. Japan Acad. Series A 72 (1996), 160-161. CMP 97:04
  • [10] Zyozyu Ohmori, On the Schur indices of certain irreducible characters of finite Chevalley groups, Hokkaido Math. J. 28 (1999), no. 1, 39–55. MR 1673478, 10.14492/hokmj/1351001076
  • [11] Udo Riese and Peter Schmid, Schur indices and Schur groups. II, J. Algebra 182 (1996), no. 1, 183–200. MR 1388863, 10.1006/jabr.1996.0167
  • [12] Peter Schmid, Schur indices and Schur groups, J. Algebra 169 (1994), no. 1, 226–247. MR 1296591, 10.1006/jabr.1994.1281
  • [13] Alexandre Turull, The Schur index of projective characters of symmetric and alternating groups, Ann. of Math. (2) 135 (1992), no. 1, 91–124. MR 1147958, 10.2307/2946564
  • [14] A. Turull, The Schur indices of the irreducible characters of the special linear groups, J. Algebra 235 (2001), 275-314.

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

Alexandre Turull
Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611
Email: turull@math.ufl.edu

DOI: https://doi.org/10.1090/S0002-9939-01-06072-5
Keywords: Brauer group, Schur index, linear groups, classical groups, characters, representations
Received by editor(s): June 23, 2000
Received by editor(s) in revised form: July 14, 2000
Published electronically: June 8, 2001
Additional Notes: The author was partially supported by a grant from the NSA
Communicated by: Stephen D. Smith
Article copyright: © Copyright 2001 American Mathematical Society