Finite subgroups of algebraic groups

Larsen, Michael; Pink, Richard

doi:10.1090/S0894-0347-2011-00695-4

Finite subgroups of algebraic groups
HTML articles powered by AMS MathViewer

by Michael J. Larsen and Richard Pink

J. Amer. Math. Soc. 24 (2011), 1105-1158

DOI: https://doi.org/10.1090/S0894-0347-2011-00695-4

Published electronically: April 28, 2011

PDF | Request permission

Abstract:

Generalizing a classical theorem of Jordan to arbitrary characteristic, we prove that every finite subgroup of $\operatorname {GL}_n$ over a field of any characteristic $p$ possesses a subgroup of bounded index which is composed of finite simple groups of Lie type in characteristic $p$, a commutative group of order prime to $p$, and a $p$-group. While this statement can be deduced from the classification of finite simple groups, our proof is self-contained and uses methods only from algebraic geometry and the theory of linear algebraic groups. We believe that our results can serve as a viable substitute for classification in a range of applications in various areas of mathematics.

References

Armand Borel, Linear algebraic groups, 2nd ed., Graduate Texts in Mathematics, vol. 126, Springer-Verlag, New York, 1991. MR 1102012, DOI 10.1007/978-1-4612-0941-6
Richard Brauer and Walter Feit, An analogue of Jordan’s theorem in characteristic $p$, Ann. of Math. (2) 84 (1966), 119–131. MR 200350, DOI 10.2307/1970514
Roger W. Carter, Simple groups of Lie type, Pure and Applied Mathematics, Vol. 28, John Wiley & Sons, London-New York-Sydney, 1972. MR 0407163
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
Michael J. Collins, On Jordan’s theorem for complex linear groups, J. Group Theory 10 (2007), no. 4, 411–423. MR 2334748, DOI 10.1515/JGT.2007.032
Michael J. Collins, Modular analogues of Jordan’s theorem for finite linear groups, J. Reine Angew. Math. 624 (2008), 143–171. MR 2456628, DOI 10.1515/CRELLE.2008.084
Demazure, M., Grothendieck, A., (Eds.), Schémas en Groupes I–III, Séminaire de Géométrie Algébrique du Bois Marie 1962/64, SGA3, Lect. Notes Math. 151–153, Berlin: Springer (1970).
Dickson, L. E., Linear groups: with an exposition of the Galois field theory, Leipzig: B. G. Teubner (1901).
Grothendieck, A., Dieudonné, J. A., Éléments de Géométrie Algébrique I, EGA1, Berlin: Springer (1971).
A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. I, Inst. Hautes Études Sci. Publ. Math. 20 (1964), 259 (French). MR 173675
Robert M. Guralnick, Small representations are completely reducible, J. Algebra 220 (1999), no. 2, 531–541. MR 1717357, DOI 10.1006/jabr.1999.7963
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157, DOI 10.1007/978-1-4757-3849-0
Gerhard Hiss, Die adjungierten Darstellungen der Chevalley-Gruppen, Arch. Math. (Basel) 42 (1984), no. 5, 408–416 (German). MR 756692, DOI 10.1007/BF01190689
G. M. D. Hogeweij, Almost-classical Lie algebras. I, II, Nederl. Akad. Wetensch. Indag. Math. 44 (1982), no. 4, 441–452, 453–460. MR 683531, DOI 10.1016/1385-7258(82)90037-3
E. Hrushovski and A. Pillay, Definable subgroups of algebraic groups over finite fields, J. Reine Angew. Math. 462 (1995), 69–91. MR 1329903
Ehud Hrushovski and Frank Wagner, Counting and dimensions, Model theory with applications to algebra and analysis. Vol. 2, London Math. Soc. Lecture Note Ser., vol. 350, Cambridge Univ. Press, Cambridge, 2008, pp. 161–176. MR 2436141, DOI 10.1017/CBO9780511735219.005
James E. Humphreys, Linear algebraic groups, Graduate Texts in Mathematics, No. 21, Springer-Verlag, New York-Heidelberg, 1975. MR 0396773, DOI 10.1007/978-1-4684-9443-3
James E. Humphreys, Conjugacy classes in semisimple algebraic groups, Mathematical Surveys and Monographs, vol. 43, American Mathematical Society, Providence, RI, 1995. MR 1343976, DOI 10.1090/surv/043
Jens Carsten Jantzen, Representations of algebraic groups, Pure and Applied Mathematics, vol. 131, Academic Press, Inc., Boston, MA, 1987. MR 899071
Jordan, C., Mémoire sur les équations differentielles linéaires à intégrale algébrique, J. für Math. 84 (1878), 89–215.
Nicholas M. Katz, Gauss sums, Kloosterman sums, and monodromy groups, Annals of Mathematics Studies, vol. 116, Princeton University Press, Princeton, NJ, 1988. MR 955052, DOI 10.1515/9781400882120
M. Kneser, Semi-simple algebraic groups, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965) Thompson, Washington, D.C., 1967, pp. 250–265. MR 0217077
G. I. Lehrer, Rational tori, semisimple orbits and the topology of hyperplane complements, Comment. Math. Helv. 67 (1992), no. 2, 226–251. MR 1161283, DOI 10.1007/BF02566498
David Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, Published for the Tata Institute of Fundamental Research, Bombay by Oxford University Press, London, 1970. MR 0282985
David Mumford, Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Band 34, Springer-Verlag, Berlin-New York, 1965. MR 0214602, DOI 10.1007/978-3-662-00095-3
Madhav V. Nori, On subgroups of $\textrm {GL}_n(\textbf {F}_p)$, Invent. Math. 88 (1987), no. 2, 257–275. MR 880952, DOI 10.1007/BF01388909
Richard Pink, The Mumford-Tate conjecture for Drinfeld-modules, Publ. Res. Inst. Math. Sci. 33 (1997), no. 3, 393–425. MR 1474696, DOI 10.2977/prims/1195145322
Richard Pink, Compact subgroups of linear algebraic groups, J. Algebra 206 (1998), no. 2, 438–504. MR 1637068, DOI 10.1006/jabr.1998.7439
Jean-Pierre Serre, Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math. 15 (1972), no. 4, 259–331 (French). MR 387283, DOI 10.1007/BF01405086
Robert Steinberg, Endomorphisms of linear algebraic groups, Memoirs of the American Mathematical Society, No. 80, American Mathematical Society, Providence, R.I., 1968. MR 0230728
John Tate, Endomorphisms of abelian varieties over finite fields, Invent. Math. 2 (1966), 134–144. MR 206004, DOI 10.1007/BF01404549
Boris Weisfeiler, Post-classification version of Jordan’s theorem on finite linear groups, Proc. Nat. Acad. Sci. U.S.A. 81 (1984), no. 16, , Phys. Sci., 5278–5279. MR 758425, DOI 10.1073/pnas.81.16.5278