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On the conjectures of J. Thompson and O. Ore


Authors: Erich W. Ellers and Nikolai Gordeev
Journal: Trans. Amer. Math. Soc. 350 (1998), 3657-3671
MSC (1991): Primary 20G15
DOI: https://doi.org/10.1090/S0002-9947-98-01953-9
MathSciNet review: 1422600
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Abstract: If $G$ is a finite simple group of Lie type over a field containing more than $8$ elements (for twisted groups $^{l} X_{n} (q^{l})$ we require $q > 8$, except for $^{2} B_{2} (q^{2})$, $^{2} G_{2} (q^{2})$, and $^{2} F_{4} (q^{2})$, where we assume $q^{2} > 8$), then $G$ is the square of some conjugacy class and consequently every element in $G$ is a commutator.


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  • [AH] Products of conjugacy classes in groups, Lecture Notes in Mathematics, no. 1112 (Z. Arad and M. Herzog, eds.), Springer Verlag, New York, 1985. MR 87h:20001
  • [B] O. Bonten, Über Kommutatoren in endlichen einfachen Gruppen, Aachener Beiträge zur Mathematik, Bd. 7, Verlag der Augustinus-Buchhandlung, Aachen, 1993.
  • [Bo] N. Bourbaki, Groupes et algèbres de Lie IV, V, VI, Hermann, Paris, 1968. MR 39:1590
  • [Br] J. L. Brenner, Covering theorems for finasigs X, Ars Combinatoria 16 (1983), 57-67. MR 85g:20021
  • [BrL] J. L. Brenner and R. J. List, Application of partition theory to groups: Covering the alternating group by products of conjugacy classes, Number Theory (J. M. De Koninck and C. Levesque, eds.), Walter de Gruyter, Berlin, New York, 1989, pp. 65-71. MR 90k:20029
  • [C1] R. W. Carter, Simple Groups of Lie Type, John Wiley & Sons, London, 1989. MR 90g:20001
  • [C2] R. W. Carter, Finite Groups of Lie Type, John Wiley & Sons, Chichester, 1993. MR 94k:20020
  • [EGI] E. W. Ellers and N. L. Gordeev, Gauss decomposition with prescribed semisimple part in classical Chevalley groups, Comm Algebra 22 (1994, no. 14), 5935-5950. MR 95m:20052
  • [EGII] E. W. Ellers and N. L. Gordeev, Gauss decomposition with prescribed semisimple part in Chevalley groups II: Exceptional cases, Comm Algebra 23 (1995, no. 8), 3085-3098. MR 96f:20064
  • [EGIII] E. W. Ellers and N. L. Gordeev, Gauss decomposition with prescribed semisimple part in Chevalley groups III: Twisted groups, Comm Algebra 24 (1996, no. 14), 4447-4475. MR 98a:20048
  • [G1] D. Gluck, Character value estimates for groups of Lie type, Pacific J. Math. 150 (1991), 279-307. MR 92k:20014
  • [G2] D. Gluck, Character value estimates for non-semisimple elements, J. Algebra 155 (1993), 221-237. MR 94b:20023
  • [G3] D. Gluck, Sharper character value estimates for groups of Lie type, J. Algebra 174 (1995), 229-266. MR 96m:20021
  • [Go] M. Goto, A theorem on compact semisimple groups, J. Math. Soc. Japan 1 (1949), 270-272. MR 11:497d
  • [Gow] R. Gow, Commutators in the symplectic group, Arch. Math. (Basel) 50 (1988), 204-209. MR 89g:20071
  • [H] C. Hsü (Xu), The commutators of the alternating group, Sci. Sinica 14 (1965), 339-342. MR 32:1241
  • [I] I. M. Isaacs, Character theory of finite groups, Academic Press, New York, 1976. MR 57:417
  • [L] T. J. Laffey, Products of matrices, Generators and relations in groups and geometries, (A. Barlotti, E. W. Ellers, P. Plaumann, K. Strambach, eds.), NATO ASI Series C, vol. 333, Kluwer Academic Publishers, Dordrecht, 1991, pp. 95-123. MR 93m:15018
  • [Le] A. Lev, Products of cyclic similarity classes in the groups $GL_{n} (F)$, Linear Algebra Appl. 202 (1994), 235-266. MR 95e:20062
  • [MSaWe] G. Malle, J. Saxl, and T. Weigel, Generation of classical groups, Geom. Dedicata 40 (1994), 85-116. MR 95c:20068
  • [NPaCl] J. Neubüser, H. Pahlings and E. Cleuvers, Each sporadic finasig $G$ has a class $C$ such that $CC=G$, Abstracts AMS 34 (1984), 6.
  • [O] O. Ore, Some remarks on commutators, Proc. Amer. Math. Soc. 272 (1951), 307-314. MR 12:671e
  • [PW] S. Pasiencier and H. C. Wang, Commutators in a complex semisimple Lie group, Proc. Amer. Math. Soc. 13 (1962), 907-913. MR 30:190
  • [R] R. Ree, Commutators in semi-simple algebraic groups, Proc. Amer. Math. Soc. 15 (1964), 457-460. MR 28:5148
  • [S] K. Shoda, Über den Kommutator der Matrizen, J. Math. Soc. Japan 3 (1951), 78-81. MR 13:425b
  • [So] A. R. Sourour, A factorization theorem for matrices, Linear and Multilinear Algebra 19 (1986, no. 2), 141-147. MR 87j:15028
  • [St] R. Steinberg, Lectures on Chevalley Groups, Yale University, 1967. MR 57:6215
  • [VWh] L. N. Vaserstein and E. Wheland, Commutators and companion matrices over rings of stable rank one, Linear Algebra Appl. 142 (1990), 263-277. MR 92a:19002
  • [Wi] J. S. Wilson, On simple pseudofinite groups, J. London Math. Soc. (2) 51 (1995), 471-490. MR 96c:20005

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Additional Information

Erich W. Ellers
Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 3G3
Email: ellers@math.utoronto.ca

Nikolai Gordeev
Affiliation: Department of Mathematics, Russian State Pedagogical University, Moijka 48, St. Petersburg, Russia 191-186
Email: algebra@ivt.rgpu.spb.ru

DOI: https://doi.org/10.1090/S0002-9947-98-01953-9
Received by editor(s): April 5, 1996
Received by editor(s) in revised form: October 10, 1996
Additional Notes: Research supported in part by NATO collaborative research grant CRG 950689
Article copyright: © Copyright 1998 American Mathematical Society

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