On linear groups over finite fields
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- by Ji Ping Zhang
- Proc. Amer. Math. Soc. 110 (1990), 53-57
- DOI: https://doi.org/10.1090/S0002-9939-1990-1028297-1
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Abstract:
Let $G$ be a finite group with an Abelian Sylow $p$-subgroup $P$ $(p > 5)$, and $F$, a finite field of characteristic $p$. Set $H = {O^{pβ}}(G)$. If $G$ has a faithful FG-module $M$ such that ${\dim _F}M < p - 2$, then one of the following is true: (a) $P$ is normal in $G$, (b) $H/Z(H) \approx { \oplus _{i \leq t}}{L_2}({p^{{n_i}}})$, where ${n_i}$ and $t$ are positive integers and $2t < p - 2$, (c) $p = 7{\text {or}}11$ and $H \approx 2.{A_7}$ or ${J_1}$, respectively, ${\dim _F}M \geq p - 4$.References
- Richard Brauer, On groups whose order contains a prime number to the first power. I, Amer. J. Math. 64 (1942), 401β420. MR 6537, DOI 10.2307/2371693
- Richard Brauer, Representations of finite groups, Lectures on Modern Mathematics, Vol. I, Wiley, New York, 1963, pp.Β 133β175. MR 0178056
- Richard Brauer and Hsio-Fu Tuan, On simple groups of finite order. I, Bull. Amer. Math. Soc. 51 (1945), 756β766. MR 15102, DOI 10.1090/S0002-9904-1945-08441-9
- Hsio-Fu Tuan, On groups whose orders contain a prime number to the first power, Ann. of Math. (2) 45 (1944), 110β140. MR 9397, DOI 10.2307/1969079
- Walter Feit, Groups with a cyclic Sylow subgroup, Nagoya Math. J. 27 (1966), 571β584. MR 199255
- Walter Feit, On finite linear groups, J. Algebra 5 (1967), 378β400. MR 207818, DOI 10.1016/0021-8693(67)90049-X
- Walter Feit, The representation theory of finite groups, North-Holland Mathematical Library, vol. 25, North-Holland Publishing Co., Amsterdam-New York, 1982. MR 661045
- Ji Ping Zhang, Complex linear groups of degree at most $p-1$, Classical groups and related topics (Beijing, 1987) Contemp. Math., vol. 82, Amer. Math. Soc., Providence, RI, 1989, pp.Β 243β254. MR 982293, DOI 10.1090/conm/082/982293 β, On linear groups of degrees at most $|P| - 1$, J. Algebra (to appear).
- Pamela A. Ferguson, Complex linear groups of degree at most $v-3$, J. Algebra 93 (1985), no.Β 2, 246β252. MR 786752, DOI 10.1016/0021-8693(85)90158-9
- H. I. Blau, Brauer trees and character degrees, Proceedings of the Symposium on Modular Representations of Finite Groups (Math. Inst., Univ. Aarhus, Aarhus, 1978) Various Publications Series, vol. 29, Aarhus Univ., Aarhus, 1978, pp.Β 10β11. MR 539268
- Vicente Landazuri and Gary M. Seitz, On the minimal degrees of projective representations of the finite Chevalley groups, J. Algebra 32 (1974), 418β443. MR 360852, DOI 10.1016/0021-8693(74)90150-1
- Roger W. Carter, Simple groups of Lie type, Pure and Applied Mathematics, Vol. 28, John Wiley & Sons, London-New York-Sydney, 1972. MR 0407163
- 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 H. F. Blichfeldt, Finite collineation groups, Univ. of Chicago Press, Chicago, 1917.
- Fletcher Gross, Automorphisms which centralize a Sylow $p$-subgroup, J. Algebra 77 (1982), no.Β 1, 202β233. MR 665174, DOI 10.1016/0021-8693(82)90287-3
Bibliographic Information
- © Copyright 1990 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 110 (1990), 53-57
- MSC: Primary 20C20
- DOI: https://doi.org/10.1090/S0002-9939-1990-1028297-1
- MathSciNet review: 1028297