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Finite quotients of the multiplicative group of a finite dimensional division algebra are solvable


Authors: Andrei S. Rapinchuk, Yoav Segev and Gary M. Seitz
Journal: J. Amer. Math. Soc. 15 (2002), 929-978
MSC (1991): Primary 16K20, 16U60; Secondary 20G15, 05C25
DOI: https://doi.org/10.1090/S0894-0347-02-00393-4
Published electronically: June 21, 2002
MathSciNet review: 1915823
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Abstract: We prove that finite quotients of the multiplicative group of a finite dimensional division algebra are solvable. Let $D$ be a finite dimensional division algebra having center $K$, and let $N\subseteq D^{\times}$ be a normal subgroup of finite index. Suppose $D^{\times}/N$ is not solvable. Then we may assume that $H:=D^{\times}/N$ is a minimal nonsolvable group (MNS group for short), i.e. a nonsolvable group all of whose proper quotients are solvable. Our proof now has two main ingredients. One ingredient is to show that the commuting graph of a finite MNS group satisfies a certain property which we denote Property $(3\frac{1}{2})$. This property includes the requirement that the diameter of the commuting graph should be $\ge 3$, but is, in fact, stronger. Another ingredient is to show that if the commuting graph of $D^{\times}/N$ has Property $(3\frac{1}{2})$, then $N$ is open with respect to a nontrivial height one valuation of $D$ (assuming without loss of generality, as we may, that $K$ is finitely generated). After establishing the openness of $N$ (when $D^{\times}/N$ is an MNS group) we apply the Nonexistence Theorem whose proof uses induction on the transcendence degree of $K$ over its prime subfield to eliminate $H$ as a possible quotient of $D^{\times}$, thereby obtaining a contradiction and proving our main result.


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

  • [1] S. Amitsur, Finite subgroups of division rings, Trans. AMS 80(1955), 361-386. MR 17:577c
  • [2] M. Aschbacher, G.M. Seitz, Involutions in Chevalley groups over fields of even order, Nagoya Math. J. 63(1976), 1-91. MR 54:10391
  • [3] J.H. Conway et al., ATLAS of finite groups, Clarendon Press, Oxford, 1985. MR 88g:20025
  • [4] H. Behr, Arithmetic groups over function fields. I. A complete characterization of finitely generated and finitely presented arithmetic subgroups of reductive algebraic groups, J. Reine Angew. Math. 495(1998), 79-118. MR 99g:20088
  • [5] N. Bourbaki, Algèbre commutative, Ch. V-VI, Masson, Paris, 1985. MR 86k:13001b
  • [6] V. Bergelson, D.B. Shapiro, Multiplicative subgroups of finite index in a ring, Proc. AMS 116(1992), 885-896. MR 93b:16001
  • [7] R.W. Carter, Conjugacy classes in the Weyl group, Compositio Math. 25(1972), 1-59. MR 47:6884
  • [8] -,    Finite Groups of Lie Type: Conjugacy Classes and Complex Characters, Wiley-Interscience, 1985. MR 87d:20060
  • [9] D. Gorenstein, R. Lyons, The local structure of finite groups of characteristic 2 type, Memoirs AMS 276 (1983), 1-731. MR 84g:20025
  • [10] S. Lang, Algebra, Addison-Wesley, 1965. MR 33:5416
  • [11] R. Lawther, M.W. Liebeck, G.M. Seitz, Fixed point ratios in actions of finite exceptional groups of Lie type, to appear, Pacific J. Math.
  • [12] M.W. Liebeck, G.M. Seitz, Reductive Subgroups of Exceptional Algebraic Groups, Memoirs AMS 580(1996), 1-111. MR 96i:20059
  • [13] G.A. Margulis, Finiteness of quotients of discrete groups, Funct. Analysis and Appl. 13(1979), 178-187. MR 80k:22006
  • [14] R. Pierce, Associative Algebras, GTM 88, Springer, 1982. MR 84c:16001
  • [15] V.P. Platonov, A.S. Rapinchuk, Algebraic Groups and Number Theory, ``Pure and Applied Mathematics'' series, N 139, Academic Press, 1993. MR 95b:11039
  • [16] -,    The multiplicative structure of division algebras over number fields and the Hasse norm principle, Proc. Steklov Inst. Math. 165(1985), 187-205. MR 85j:11162
  • [17] G. Prasad, Strong approximation for semi-simple groups over function fields, Ann. Math. 105(1977), 553-572. MR 56:2921
  • [18] M.S. Raghunathan, On the group of norm 1 elements in a division algebra, Math. Ann. 279(1988), 457-484. MR 89g:11111
  • [19] A.S. Rapinchuk, A. Potapchik, Normal subgroups of $SL_{1,D}$ and the classification of finite simple groups, Proc. Indian Acad. Sci. 106(1996), 329-368. MR 98i:20049
  • [20] A.S. Rapinchuk, Y. Segev, Valuation-like maps and the congruence subgroup property, Invent. Math. 144(2001), 571-607. MR 2002e:16027
  • [21] L. Rowen, Y. Segev, The finite quotients of the multiplicative group of a division algebra of degree 3 are solvable, Israel J. of Math. 111(1999), 373-380. MR 2000g:16029
  • [22] -,    The multiplicative group of a division algebra of degree 5 and Wedderburn's Factorization Theorem, Contemp. Math. 259(2000), 475-486. MR 2001g:16037
  • [23] Y. Segev, On finite homomorphic images of the multiplicative group of a division algebra, Ann. Math. 149(1999), 219-251. MR 2000e:16022
  • [24] -,    Some applications of Wedderburn's factorization theorem, Bull. Austral. Math. Soc. 59(1999), 105-110. MR 99k:16068
  • [25] -,    The commuting graph of minimal nonsolvable groups, Geom. Ded. 88(2001), 55-66.
  • [26] Y. Segev, G.M. Seitz, Anisotropic groups of type $A_n$ and the commuting graph of finite simple groups, Pacific J. Math. 202(2002), 125-226. CMP 2002:08
  • [27] A. Seress, The minimal base size of primitive solvable permutation groups, J. LMS (2) 53(1996), no. 2, 243-255. MR 96k:20003
  • [28] T.A. Springer, R. Steinberg, Conjugacy Classes, Springer Lecture Notes 131 (eds: A. Borel, et al.), Springer, Berlin, 1970. MR 42:3091
  • [29] R. Steinberg, Lectures on Chevalley Groups, Yale University Lecture Notes, 1967. MR 57:6215
  • [30] M. Suzuki, On a class of doubly transitive groups, Ann. Math. 75 (1962), 105-145. MR 25:112
  • [31] J. Tits, Algebraic and abstract simple groups, Ann. Math. 80(1964), no. 2, 313-329. MR 29:2259
  • [32] -,    Groupes de Whitehead de groupes algebriques simples sur un corps (d'apres V.P. Platonov et al.), Sem. Bourbaki, 1977, exp. 505. Lecture Notes in Math. 677(1978), 218-236. MR 80d:12008
  • [33] G. Turnwald, Multiplicative subgroups of finite index in rings, Proc. AMS 120(1994), 377-381. MR 94e:12002
  • [34] A.R. Wadsworth, Extending valuations to finite-dimensional division algebras, Proc. AMS 98(1986), 20-22. MR 87i:16025

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Additional Information

Andrei S. Rapinchuk
Affiliation: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22904
Email: asr3x@weyl.math.virginia.edu

Yoav Segev
Affiliation: Department of Mathematics, Ben-Gurion University, Beer-Sheva 84105, Israel
Email: yoavs@math.bgu.ac.il

Gary M. Seitz
Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1226
Email: seitz@math.uoregon.edu

DOI: https://doi.org/10.1090/S0894-0347-02-00393-4
Keywords: Division algebra, multiplicative group, finite homomorphic images, valuations
Received by editor(s): February 28, 2001
Received by editor(s) in revised form: January 24, 2002
Published electronically: June 21, 2002
Additional Notes: The first author was partially supported by grants from the NSF and by BSF grant no. 97-00042
The second author was partially supported by BSF grant no. 97-00042. Portions of this work were written while the author visited the Forschungsinstitut für Mathematik ETH, Zurich, in the summer of 2000, and the author gratefully acknowledges the hospitality and support.
The third author was partially supported by grants from the NSF and by BSF grant no. 97-00042.
Article copyright: © Copyright 2002 American Mathematical Society

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