Finite linear groups and theorems of Minkowski and Schur
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- by Walter Feit PDF
- Proc. Amer. Math. Soc. 125 (1997), 1259-1262 Request permission
Abstract:
Let $G$ be a finite group with a faithful rational valued character of degree $n$. A theorem of I. Schur gives a bound for the order of $G$ in terms of $n$, generalizing an earlier result of H. Minkowski who showed that the same bound holds if $G\subseteq GL(n,\mathbf {Q})$. This note contains strengthened versions of these results which in particular show that a $2$-subgroup of $GL(n,\mathbf {Q})$ of maximum possible order contains a reflection.References
- N. Bourbaki, Éléments de mathématique. Fasc. XXXVII. Groupes et algèbres de Lie. Chapitre II: Algèbres de Lie libres. Chapitre III: Groupes de Lie, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1349, Hermann, Paris, 1972. MR 0573068
- Walter Feit, Characters of finite groups, W. A. Benjamin, Inc., New York-Amsterdam, 1967. MR 0219636
- Issai Schur, Gesammelte Abhandlungen. Band I, Springer-Verlag, Berlin-New York, 1973 (German). Herausgegeben von Alfred Brauer und Hans Rohrbach. MR 0462891
Additional Information
- Walter Feit
- Affiliation: Department of Mathematics, Yale University, P.O. Box 208283, New Haven, Connecticut 06520-8283
- Email: feit@math.yale.edu
- Received by editor(s): November 1, 1995
- Communicated by: Ronald M. Solomon
- © Copyright 1997 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 125 (1997), 1259-1262
- MSC (1991): Primary 20C15; Secondary 20H20
- DOI: https://doi.org/10.1090/S0002-9939-97-03801-X
- MathSciNet review: 1376761