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The Commutators of Classical Groups

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In his seminal paper, half a century ago, Hyman Bass established commutator formulas for a (stable) general linear group, which were the key step in defining the group K 1. Namely, he proved that for an associative ring A with identity,

$$ E(A)=\left[E(A),E(A)\right]=\left[\mathrm{GL}(A),\mathrm{GL}(A)\right], $$

where GL(A) is the stable general linear group and E(A) is its elementary subgroup. Since then, various commutator formulas have been studied in stable and non-stable settings for classical groups, algebraic groups, and their analogs, and mostly in relation to subnormal subgroups of these groups. The basic classical theorems and methods developed for their proofs are associated with the names of the heroes of classical algebraic K-theory: Bak, Quillen, Milnor, Suslin, Swan, Vaserstein, and others.

One of the dominant techniques in establishing commutator type results is localization. In the present paper, some recent applications of localization methods to the study (higher/relative) commutators in the groups of points of algebraic and algebraic-like groups, such as general linear groups GL(n,A), unitary groups GU(2n,A, Λ), and Chevalley groups G(Φ,A), are described. Some auxiliary results and corollaries of the main results are also stated.

The paper provides a general overview of the subject and covers the current activities. It contains complete proofs borrowed from our previous papers and expositions of several main results to give the reader a self-contained source.

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References

  1. E. Abe, “Whitehead groups of Chevalley groups over polynomial rings,” Commun. Algebra, 11, No. 12, 1271–1308 (1983).

    Article  MathSciNet  MATH  Google Scholar 

  2. E. Abe, “Chevalley groups over commutative rings,” in: Proceedings of the conference on Radical Theory, Sendai (1988), pp. 1–23.

  3. E. Abe, “Normal subgroups of Chevalley groups over commutative rings,” Contemp. Math., 83, 1–17 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  4. M. Akhavan-Malayeri, “Writing certain commutators as products of cubes in free groups,” J. Pure Appl. Algebra, 177, No. 1, 1–4 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  5. M. Akhavan-Malayeri, “Writing commutators of commutators as products of cubes in groups,” Commun. Algebra, 37, 2142–2144 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  6. H. Apte and A. Stepanov, “Local-global principle for congruence subgroups of Chevalley groups,” Central Europ. J. Math., 12, No. 6, 801–812 (2014).

    MathSciNet  MATH  Google Scholar 

  7. A. Bak, “The stable structure of quadratic modules,” Thesis, Columbia University (1969).

  8. A. Bak, K-Theory of Forms, Princeton University Press. Princeton (1981).

    MATH  Google Scholar 

  9. A. Bak, “Subgroups of the general linear group normalized by relative elementary groups,” Lecture Notes Math., 967, 1–22 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  10. A. Bak, “Non-abelian K-theory: the nilpotent class of K1 and general stability,” K-Theory, 4, 363–397 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  11. A. Bak, R. Basu, and R. A. Rao, “Local-global principle for transvection groups,” Proc. Amer. Math. Soc., 138, No. 4, 1191–1204 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  12. A. Bak, R. Hazrat, and N. A. Vavilov, “Localization-completion strikes again: relative K1 is nilpotent by abelian,” J. Pure Appl. Algebra, 213, 1075–1085 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  13. A. Bak and A. Stepanov, “Dimension theory and nonstable K-theory for net groups,” Rend. Sem. Mat. Univ. Padova, 106, 207–253 (2001).

    MathSciNet  MATH  Google Scholar 

  14. A. Bak and N. A. Vavilov, “Normality for elementary subgroup functors,” Math. Proc. Cambridge Phil. Soc., 118, No. 1, 35–47 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  15. A. Bak and N. A. Vavilov, “Structure of hyperbolic unitary groups. I. Elementary subgroups,” Algebra Colloq. 7, No. 2, 159–196 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  16. H. Bass, “K-theory and stable algebra,” Inst. Hautes Études Sci. Publ. Math., No. 22, 5–60 (1964).

  17. H. Bass, J. Milnor, and J.-P. Serre, “Solution of the congruence subgroup problem for SL n (n ≥ 3) and Sp2n (n ≥ 2),” Inst. Hautes Études Sci. Publ. Math., No. 33, 59–133 (1967).

  18. H. Bass, “Unitary algebraic K-theory,” Lecture Notes Math., 343, 57–265 (1973).

    Article  MathSciNet  MATH  Google Scholar 

  19. H. Bass, Algebraic K-theory, Benjamin, New York (1968).

    MATH  Google Scholar 

  20. R. Basu, “Topics in Classical Algebraic K-theory,” Ph. D. Thesis, Tata Institute of Fundamental Research, Mumbai (2007).

  21. R. Basu, “Local-global principle for general quadratic and general hermitian groups and the nilpotence of KH1,” arXiv:1412.3631v1 (2014).

  22. R. Basu, R. A. Rao, and R. Khanna, “On Quillen’s local global principle,” Contemp. Math., 390, 17–30 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  23. Z. Borewicz and N. A. Vavilov, “The distribution of subgroups in the full linear group over a commutative ring,” Proc. Steklov Institute Math., 3, 27–46 (1985).

    Google Scholar 

  24. P. Chattopadhya and R. A. Rao, “Excision and elementary symplectic action,” preprint, 1–14 (2012).

  25. R. K. Dennis and L. N. Vaserstein, “On a question of M. Newman on the number of commutators,” J. Algebra, 118, 150–161 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  26. R. K. Dennis and L. N. Vaserstein, “Commutators in linear groups,” K-Theory, 2, 761–767 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  27. E. Ellers and N. Gordeev, “On the conjectures of J. Thompson and O. Ore,” Trans. Amer. Math. Soc., 350, 3657–3671 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  28. D. Estes and J. Ohm, “Stable range in commutative rings,” J. Algebra, 7, No. 3, 343–362 (1967).

    Article  MathSciNet  MATH  Google Scholar 

  29. S. C. Geller and C. A. Weibel, “K2 measures excision for K1,” Proc. Amer. Math. Soc., 80, No. 1, 1–9 (1980).

    MathSciNet  MATH  Google Scholar 

  30. S. C. Geller and C. A. Weibel, “K1(A,B, I),” J. Reine Angew. Math., 342, 12–34 (1983).

    MathSciNet  MATH  Google Scholar 

  31. S. C. Geller and C. A. Weibel, “Subroups of elementary and Steinberg groups of congruence level I 2,” J. Pure Appl. Algebra, 35, 123–132 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  32. S. C. Geller and C. A. Weibel, “K1(A,B, I), II,” K-Theory, 2, No. 6, 753–760 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  33. V. N. Gerasimov, “The group of units of the free product of rings,” Mat. Sb., 134, 42–65 (1987).

    MATH  Google Scholar 

  34. R. M. Guralnick and G. Malle, “Products of conjugacy classes and fixed point spaces,” J. Amer. Math. Soc., 25, No. 1, 77–121 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  35. G. Habdank, “A classification of subgroups of Λ-quadratic groups normalized by relative elementary subgroups,” Dissertation, Universität Bielefeld (1987).

  36. G. Habdank, “A classification of subgroups of Λ-quadratic groups normalized by relative elementary subgroups,” Adv. Math., 110, No. 2, 191–233 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  37. A. J. Hahn and O. T. O’Meara, The Classical Groups and K-Theory, Springer, Berlin etc. (1989).

  38. R. Hazrat, “Dimension theory and non-stable K1 of quadratic module,” K-Theory, 27, 293–327 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  39. R. Hazrat, V. Petrov, and N. Vavilov, “Relative subgroups in Chevalley groups,” J. K-Theory, 5, 603–618 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  40. R. Hazrat, A. Stepanov, N. Vavilov, and Z. Zhang, “The yoga of commutators,” Zap. Nauchn. Semin. POMI, 287, 53–82 (2011).

    MathSciNet  MATH  Google Scholar 

  41. R. Hazrat, A. Stepanov, N. Vavilov, and Z. Zhang, “Commutators width in Chevalley groups,” Note di Matematica, 33, No. 1, 139–170 (2013).

    MathSciNet  MATH  Google Scholar 

  42. R. Hazrat, A. Stepanov, N. Vavilov, and Z. Zhang, “Multiple commutator formula. II” (2012).

  43. R. Hazrat and N. Vavilov, “K1 of Chevalley groups are nilpotent,” J. Pure Appl. Algebra, 179, 99–116 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  44. R. Hazrat and N. Vavilov, “Bak’s work on K-theory of rings (with an appendix by Max Karoubi), K-Theory 4, No. 1, 1–65 (2009).

  45. R. Hazrat, N. Vavilov, and Z. Zhang, “Relative commutator calculus in unitary groups, and applications,” J. Algebra, 343, 107–137 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  46. R. Hazrat, N. Vavilov, and Z. Zhang, “Relative commutator calculus in Chevalley groups, and applications,” J. Algebra, 385, 262–293 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  47. R. Hazrat, N. Vavilov, and Z. Zhang, “Multiple commutator formulas for unitary groups,” Israel J. Math., (to appear).

  48. R. Hazrat, N. Vavilov, and Z. Zhang, “Generation of relative commutator subgroups in Chevalley groups,” Proc. Edinburgh Math. Soc., 59, 393–410 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  49. R. Hazrat and Z. Zhang, “Generalized commutator formula,” Commun. Algebra, 39, No. 4, 1441–1454 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  50. R. Hazrat and Z. Zhang, “Multiple commutator formula,” Israel J. Math., 195, 481–505 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  51. D. A. Jackson, “Basic commutator in weights six and seven as relators,” Commun. Algebra, 36, 2905–2909 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  52. D. A. Jackson, A. M. Gaglione, and D. Spellman, “Basic commutator as relators,” J. Group Theory, 5, 351–363 (2001).

    MathSciNet  MATH  Google Scholar 

  53. D. A. Jackson, A. M. Gaglione, and D. Spellman, “Weight five basic commutator as relators,” Contemp. Math., 511, 39–81 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  54. W. van der Kallen, “Another presentation for Steinberg groups,” Indag. Math., 39, No. 4, 304–312 (1977).

    Article  MathSciNet  MATH  Google Scholar 

  55. W. van der Kallen, “SL3(ℂ[x]) does not have bounded word length,” Lecture Notes Math., 966, 357–361 (1982).

    Article  MATH  Google Scholar 

  56. W. van der Kallen, “A module structure on certain orbit sets of unimodular rows,” J. Pure Appl. Algebra, 57, No. 3, 281–316 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  57. W. van der Kallen, B. Magurn, and L. Vaserstein, “Absolute stable rank and Witt cancellation for non-commutative rings,” Invent. Math., 91, 525–542 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  58. L.-C. Kappe and R. F. Morse, “On commutators in groups,” in: Groups St. Andrews, 2005, Vol. II, Cambridge Univ. Press, Cambridge (2007), pp. 531–558.

  59. S. Khlebutin, “Elementary subgroups of linear groups over rings,” PhD. thesis, Moscow State Univ. (1987).

  60. M.-A. Knus, Quadratic and Hermitian Forms over Rings, Springer Verlag, Berlin etc. (1991).

  61. V. I. Kopeiko, “The stabilization of symplectic groups over a polynomial ring,” Math. USSR Sb., 34, 655–669 (1978).

    Article  Google Scholar 

  62. N. Kumar and R. A. Rao, “Quillen–Suslin theory for a structure theorem for the elementary symplectic group,” Preprint (2012).

  63. Tsit-Yuen Lam, Serre’s Problem on Projective Modules, Springer Verlag, Berlin (2006).

  64. M. Larsen and A. Shalev, “Word maps and Waring type problems,” J. Amer. Math. Soc.. 22, 437–466 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  65. M. Larsen, A. Shalev, and Pham Huu Tiep, “The Waring problem for finite simple groups,” Ann. Math., 174, 1885–1950 (2011).

  66. A. V. Lavrenov, “The unitary Steinberg group is centrally closed,” St. Petersburg Math. J., 24, No. 5, 783–794 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  67. F. Li, “The structure of symplectic group over arbitrary commutative rings,” Acta Math. Sinica, 3, No. 3, 247–255 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  68. F. Li, “The structure of orthogonal groups over arbitrary commutative rings,” Chinese Ann. Math. Ser. B, 10, No. 3, 341–350 (1989).

    MathSciNet  MATH  Google Scholar 

  69. F. Li and M. Liu, “Generalized sandwich theorem,” K-Theory, 1, 171–184 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  70. M. Liebeck, E. A. O’Brien, A. Shalev, and Pham Huu Tiep, “The Ore conjecture,” J. Europ. Math. Soc., 12, 939–1008 (2010).

  71. M. Liebeck, E. A. O’Brien, A. Shalev, and Pham Huu Tiep, “Commutators in finite quasisimple groups,” Bull. London Math. Soc., 43, 1079–1092 (2011).

  72. M. Liebeck, E. A. O’Brien, A. Shalev, and Pham Huu Tiep, “Products of squares in finite simple groups,” Proc. Amer. Math. Soc., 43, No. 6, 1079–1092 (2012).

  73. A. Yu. Luzgarev, “Overgroups of E(F 4,R) in G(E 6,R),” St. Petersburg J. Math., 20, No. 5, 148–185 (2008).

    MathSciNet  Google Scholar 

  74. A. Yu. Luzgarev and A. K. Stavrova, “Elementary subgroups of isotropic reductive groups are perfect,” St. Petersburg Math. J., 24, No. 5, 881–890 (2012).

    Article  MATH  Google Scholar 

  75. A. W. Mason, “A note on subgroups of GL(n,A) which are generated by commutators,” J. London Math. Soc., 11, 509–512 (1974).

    MathSciNet  MATH  Google Scholar 

  76. A. W. Mason, “On subgroups of GL(n,A) which are generated by commutators. II,” J. Reine Angew. Math., 322, 118–135 (1981).

    MathSciNet  MATH  Google Scholar 

  77. A. W. Mason, “A further note on subgroups of GL(n,A) which are generated by commutators,” Arch. Math., 37, No. 5, 401–405 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  78. A. W. Mason and W. W. Stothers, “On subgroups of GL(n,A) which are generated by commutators,” Invent. Math., 23, 327–346 (1974).

    Article  MathSciNet  MATH  Google Scholar 

  79. H. Matsumoto, “Sur les sous-groupes arithmétiques des groupes semi-simples déployés,” Ann. Sci. École Norm. Sup., (4) 2, 1–62 (1969).

  80. J. Milnor, “Algebraic K-theory and quadratic forms,” Invent. Math., 9, 318–344 (1970).

    Article  MathSciNet  MATH  Google Scholar 

  81. J. Milnor, Introduction to algebraic K-theory, Princeton Univ. Press, Princeton, N. J. (1971).

    MATH  Google Scholar 

  82. C. Moore, “Group extensions of p-adic and adelic linear groups,” Publ. Math. Inst. Hautes Études Sci., 35, 157–222 (1968).

    Article  MathSciNet  MATH  Google Scholar 

  83. D. W. Morris, “Bounded generation of SL(n,A) (after D. Carter, G. Keller, and E. Paige),” New York J. Math., 13, 383–421 (2008).

    MathSciNet  MATH  Google Scholar 

  84. V. A. Petrov, “Overgroups of unitary groups,” K-Theory, 29, 147–174 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  85. V. A. Petrov, “Odd unitary groups,” J. Math. Sci., 130, No. 3, 4752–4766 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  86. V. A. Petrov, “Overgroups of classical groups,” Doktorarbeit Univ. St.-Petersburg, 1–129 (2005).

  87. V. A. Petrov and A. K. Stavrova, “Elementary subgroups of isotropic reductive groups,” St. Petersburg Math. J., 20, No. 3, 160–188 (2008).

    MathSciNet  MATH  Google Scholar 

  88. D. Quillen, “Projective modules over polynomial rings,” Invent. Math., 36, 166-172 (1976).

    Article  MathSciNet  MATH  Google Scholar 

  89. D. Quillen, “Higher algebraic K-theory. I,” in: Lecture Notes Math., 341, Springer, Berlin (1973), pp. 85–147.

  90. J. Rosenberg, Algebraic K-Theory and its Applications, Springer-Verlag, New York (1994).

    Book  MATH  Google Scholar 

  91. Sh. Rosset, “The higher lower central series,” Israel J. Math., 73, No. 3, 257–279 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  92. A. Sivatski and A. Stepanov, “On the word length of commutators in GL n (R),” K-Theory, 17, 295–302 (1999).

  93. A. Shalev, “Commutators, words, conjugacy classes and character methods,” Turk. J. Math., 31, 131–148 (2007).

    MathSciNet  MATH  Google Scholar 

  94. A. Shalev, “Word maps, conjugacy classes, and a noncommutative Waring-type theorem,” Ann. Math., 170, No. 3, 1383–1416 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  95. A. Smolensky, B. Sury, and N. A. Vavilov, “Gauss decomposition for Chevalley groups revisited,” Intern. J. Group Theory, 1, No. 1, 3–16 (2012).

    MathSciNet  MATH  Google Scholar 

  96. A. Stavrova, “Homotopy invariance of non-stable K1-functors,” J. K-Theory, 13, 199–248 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  97. M. R. Stein, “Generators, relations and coverings of Chevalley groups over commutative rings,” Amer. J. Math., 93, No. 4, 965–1004 (1971).

    Article  MathSciNet  MATH  Google Scholar 

  98. A. V. Stepanov, “Structure of Chevalley groups over rings via universal localization,” J. Algebra, 450, 522–548 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  99. A. V. Stepanov and N. A. Vavilov, “Decomposition of transvections: a theme with variations,” K-Theory, 19, 109–153 (2000).

  100. A. V. Stepanov and N. A. Vavilov, “On the length of commutators in Chevalley groups,” Israel J. Math., 185, 253–276 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  101. A. V. Stepanov, N. A. Vavilov, and H. You, “Overgroups of semi-simple subgroups: localization approach,” Preprint (2012).

  102. A. Stepanov, “Nonabelian K-theory for Chevalley groups over rings,” J. Math. Sci., 209, No. 4, 645–656 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  103. R. Steinberg, “Générateurs, rélations et revêtements des groupes algébriques,” in: Colloque Théorie des Groupes Algébriques (Bruxelles, 1962), Guthier–Villar, Paris (1962), pp. 113–127.

  104. R. Steinberg, Lectures on Chevalley groups Yale, University (1967).

  105. A. A. Suslin, “The structure of the special linear group over polynomial rings,” Math. USSR Izv., 11, No. 2, 235–253 (1977).

    Article  MATH  Google Scholar 

  106. A. A. Suslin and V. I. Kopeiko, “Quadratic modules and orthogonal groups over polynomial rings,” J. Sov. Math., 20, No. 6, 2665–2691 (1982).

    Article  MATH  Google Scholar 

  107. K. Suzuki, “Normality of the elementary subgroups of twisted Chevalley groups over commutative rings,” J. Algebra, 175, No. 3, 526–536 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  108. R. G. Swan, “Excision in algebraic K-theory,” J. Pure Appl. Algebra, 1, No. 3, 221–252 (1971).

    Article  MathSciNet  MATH  Google Scholar 

  109. G. Taddei, Schémas de Chevalley–Demazure, fonctions représentatives et théorème de normalité, Thèse, Univ. de Genève (1985).

  110. G. Taddei, “Normalité des groupes élémentaire dans les groupes de Chevalley sur un anneau,” Contemp. Math., 55, No. 2, 693–710 (1986).

    Article  MATH  Google Scholar 

  111. G. Tang, “Hermitian groups and K-theory,” K-Theory, 13, No. 3, 209–267 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  112. J. Tits, “Systèmes générateurs de groupes de congruences,” C. R. Acad. Sci. Paris, Sér A, 283, 693–695 (1976).

    MathSciNet  MATH  Google Scholar 

  113. M. S. Tulenbaev, “The Steinberg group of a polynomial ring,” Math. USSR Sb., 45, No. 1, 139–154 (1983).

    Article  MathSciNet  MATH  Google Scholar 

  114. M. S. Tulenbaev, “The Schur multiplier of the group of elementary matrices of finite order,” J. Sov. Math., 17, No. 4, 2062–2067 (1981).

    Article  MATH  Google Scholar 

  115. L. N. Vaserstein, “On the normal subgroups of the GL n of a ring,” Lect. Notes Math., 854, 454–465 (1981).

    MathSciNet  Google Scholar 

  116. L. N. Vaserstein, “The subnormal structure of general linear groups,” Math. Proc. Cambridge Phil. Soc., 99, 425–431 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  117. L. N. Vaserstein, “On normal subgroups of Chevalley groups over commutative rings,” Tôhoku Math. J., 36, No. 5, 219–230 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  118. L. N. Vaserstein, Normal subgroups of orthogonal groups over commutative rings. Amer. J. Math. 110, No. 5, 955–973 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  119. L. N. Vaserstein, “Normal subgroups of symplectic groups over rings,” K-Theory, 2, No. 5, 647–673 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  120. L. N. Vaserstein, “The subnormal structure of general linear groups over rings,” Math. Proc. Cambridge Phil. Soc., 108, No. 2, 219–229 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  121. L. N. Vaserstein and A. A. Suslin, “Serre’s problem on projective modules over polynomial rings, and algebraic K-theory,” Math. USSR Izv., 10, 937–1001 (1978).

    Article  MATH  Google Scholar 

  122. L. N. Vaserstein and H. You, “Normal subgroups of classical groups over rings,” J. Pure Appl. Algebra, 105, No. 1, 93–106 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  123. N. A. Vavilov, “A note on the subnormal structure of general linear groups,” Math. Proc. Cambridge Phil. Soc., 107, No. 2, 193–196 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  124. N. A. Vavilov, “Structure of Chevalley groups over commutative rings,” in: Proc. Conf. Non-associative algebras and related topics (Hiroshima – 1990), World Sci. Publ., London et al., (1991), pp. 219–335.

  125. N. A. Vavilov, “A third look at weight diagrams,” Rend. Sem. Mat. Univ. Padova, 104, No. 1, 201–250 (2000).

    MathSciNet  MATH  Google Scholar 

  126. N. A. Vavilov, A. Luzgarev, and A. Stepanov, “Calculations in exceptional groups over rings,” J. Math. Sci., 373, 48–72 (2009).

    MathSciNet  MATH  Google Scholar 

  127. N. A. Vavilov and V. A. Petrov, “Overgroups of Ep(n,R),” St. Petersburg J. Math., 15, No. 4, 515–543 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  128. N. A. Vavilov and E. B. Plotkin, “Chevalley groups over commutative rings. I. Elementary calculations,” Acta Applicandae Math., 45, 73–115 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  129. N. A. Vavilov and A. V. Stepanov, “Standard commutator formula,” Vestnik St. Petersburg Univ., Ser. 1, 41, No. 1, 5–8 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  130. N. A. Vavilov and A. V. Stepanov, “Overgroups of semi-simple groups,” Vestnik Samara State Univ., Ser. Nat. Sci., No. 3, 51–95 (2008).

  131. N. A. Vavilov and A. V. Stepanov, “Standard commutator formulae, revisited,” Vestnik St. Petersburg State Univ., Ser.1, 43, No. 1, 12–17 (2010).

    MATH  Google Scholar 

  132. N. A. Vavilov and A. V. Stepanov, “Linear groups over general rings I. Generalities,” J. Math. Sci., 188, No. 5, 490–550 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  133. N. A. Vavilov and Z. Zhang, “Subnormal subgroups of Chevalley groups. I. Cases E6 and E7,” Preprint (2015).

  134. T. Vorst, “Polynomial extensions and excision K1,” Math. Ann., 244, 193–204 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  135. M. Wendt, “\( \mathbb{A} \) 1-homotopy of Chevalley groups,” J. K-Theory, 5, No. 2, 245–287 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  136. M. Wendt, “On homotopy invariance for homology of rank two groups,” J. Pure Appl. Algebra, 216, 2291–2301 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  137. H. You, “On the solution to a question of D. G. James,” J. Northeast Normal Univ., No. 2, 39–44 (1982).

  138. H. You, “On subgroups of Chevalley groups which are generated by commutators,” J. Northeast Normal Univ., No. 2, 9–13 (1992).

  139. H. You, “Subgroups of classical groups normalised by relative elementary groups,” J. Pure Appl. Algebra, 216, 1040–1051 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  140. Z. Zhang, “Lower K-theory of unitary groups,” Doktorarbeit Univ. Belfast, 1–67 (2007).

  141. Z. Zhang, “Stable sandwich classification theorem for classical-like groups,” Math. Proc. Cambridge Phil. Soc., 143, No. 3, 607–619 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  142. Z. Zhang, “Subnormal structure of non-stable unitary groups over rings,” J. Pure Appl. Algebra, 214, 622–628 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  143. D. L. Costa and G. E. Keller, “Radix redux: normal subgroups of symplectic groups,” J. Reine Angew. Math., 427, 51–105 (1992).

    MathSciNet  MATH  Google Scholar 

  144. D. L. Costa and G. E. Keller, “On the normal suggroups of G2(A),” Trans. Amer. Math. Soc., 351, 5051–5088 (1999).

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to R. Hazrat.

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Published in Zapiski Nauchnykh Seminarov POMI, Vol. 443, 2016, pp. 151–221.

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Hazrat, R., Vavilov, N. & Zhang, Z. The Commutators of Classical Groups. J Math Sci 222, 466–515 (2017). https://doi.org/10.1007/s10958-017-3318-3

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