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Bak's work on the K-theory of rings

  • Roozbeh Hazrat (a1) and Nikolai Vavilov (a2)


This paper studies the work of Bak in algebra and (lower) algebraic K-theory and some later developments stimulated by them. We present an overview of his work in these areas, describe the setup and problems as well as the methods he introduced to attack these problems and state some of the crucial theorems. The aim is to analyse in detail some of his methods which are important and promising for further work in the subject. Among the topics covered are, unitary/general quadratic groups over form rings, structure theory and stability for such groups, quadratic K2 and the quadratic Steinberg groups, nonstable K-theory and localisation-completion, intermediate subgroups, congruence subgroup problem, dimension theory and surgery theory.



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1.Aravire, Roberto; Baeza, Ricardo, Milnor's K-theory and quadratic forms over fields of characteristic two. Comm. Algebra 20 (1992), no. 4, 10871107.
2.Artin, Emil, Geometric Algebra. Wiley Classics Library, 1988, reprint of the 1957 original.
3.Aschbacher, Michael; Some multilinear forms with large isometry groups. Geom. dedic. 25 (1988), 417465.
4.Atiyah, Michael; K-theory. Notes by D.W. Anderson. Second edition. Advanced Book Classics. Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1989.
5.Baeza, Ricardo, Some algebraic aspects of quadratic forms over fields of characteristic two. Proc. Conf Quadratic Forms and Related Topics (Baton Rouge–2001), Doc.Math. 2001, Extra Volume, 4963.
6.Bak, Anthony; The stable structure of quadratic modules, Thesis Columbia Univ., 1969
7.Bak, Anthony, On modules with quadratic forms, Lecture Notes Math. 108 (1969), 5566.
8.Bak, Anthony; The computation of surgery groups of odd torsion groups; Bull. Amer. Math. Soc. 80 (1976), no. 6, 11131116.
9.Bak, Anthony; Odd dimension surgery groups of odd torsion groups vanish. Topology 14 (1975), no. 4, 367374.
10.Bak, Anthony; The computation of surgery groups of finite groups with abelian 2-hyperelementary subgroups. Algebraic K-theory (Proc. Conf., Northwestern Univ., Evanston, Ill., 1976), 384–409. Lecture Notes in Math., Vol. 551, 1976, Springer, Berlin.
11.Bak, Anthony; Grothendieck groups of modules and forms over commutative rings; Amer. J. Math., 99 (1977); no. 1, 107120.
12.Bak, Anthony; The computation of even dimension surgery groups of odd torsion groups; Comm. Algebra. 6 (1978), no. 14, 13931458.
13.Bak, Anthony; Arf theorem for trace Noetherian and other rings. J. Pure Appl. Algebra 14 (1979), no. 1, 120.
14.Bak, Anthony, K-theory of Forms. Annals of Mathematics Studies, 98. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1981
15.Bak, Anthony; Le problème des sous-groupes de congruence et le problème métaplectique pour les groupes classiques de rang > 1. C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 5, 307310.
16.Bak, Anthony, Subgroups of the general linear group normalized by relative elementary subgroup. Algebraic K-theory, Part II (Oberwolfach, 1980), 122. Lecture Notes in Math., Vol. 967, 1982, Springer, Berlin.
17.Bak, Anthony, A norm theorem for K 2 of global fields. Algebraic topology (Aarhus, 1982), 17. Lecture Notes in Math., Vol. 1051, 1982, Springer, Berlin.
18.Bak, Anthony, Nonabelian K-theory: the nilpotent class of K 1 and general stability. K-Theory 4 (1991), no. 4, 363397.
19.Bak, Anthony, Lectures on dimension theory, group valued functors, and nonstable K-theory, Buenos Aires (1995), Preprint
20.Bak, Anthony; Induction for finite groups revisited J. Pure Appl. Algebra 104 (1995), no. 3, 235241.
21.Bak, Anthony; Global actions: an algebraic double of a topological space. Uspehi Mat. Nauk, (1997), no. 5, 71112 (Russian, English transl. Russian Math. Surveys 52 (1997), no. 5, 955–996).
22.Bak, Anthony; Topological methods in Algebra. Rings, Hopf algebras, and Brauer groups (Antwerp/Brussels, 1996), 4354, Lect. Notes Pure Appl. Math., 967, 1998, Marcel Dekker, N.Y.
23.Bak, Anthony; Kolster, Manfred; The computation of odd-dimensional projective surgery groups of finite groups. Topology 21 (1982), 3563.
24.Bak, Anthony; Morimoto, Masaharu; Cancellation over rings of dimension ≤ 1. Bull. Soc. Math. Belg., Sér A 45 (1993), no. 1–2, 2937.
25.Bak, Anthony; Morimoto, Masaharu; Equivariant surgery and applications. Topology Hawaii (Honolulu, 1990) World Sci., 1992, 1325.
26.Bak, Anthony; Morimoto, Masaharu; K-theoretical groups with positioning map and equivariant surgery. Proc. Japan. Acad. Scu, Ser. A Math. Sci. 70 (1994), no. 1, 611.
27.Bak, Anthony; Morimoto, Masaharu; Equivariant surgery with middle-dimensional singular sets. I. Forum Math. 8 (1996), no. 3, 267302.
28.Bak, Anthony; Morimoto, Masaharu; The dimension of spheres with smooth one fixed point actions. Forum Math. 17 (2005) no. 2, 199216.
29.Bak, Anthony; Morimoto, Masaharu; Equivariant Intersection Theory and Surgery Theory for Manifolds with Middle Dimensional Singular Sets, J. K-Theory 2 (2008), no. 3, 507600.
30.Bak, Anthony; Muranov, Yuri; Splitting along submanifolds, and -spectra. Sovrem. Mat. Prilozh., no. 1, 2003, 318 (Russian, English transl. J. Math. Sci. 123 (2004), no. 4, 4169–4184).
31.Bak, Anthony; Petrov, Viktor; Tang, Guoping; Stability for quadratic K 1, K-Theory 30 (2003), no. 1, 111.
32.Bak, Anthony; Rehmann, Ulf; Le problème des sous-groupes de congruence dans SLn≥2 sur un corps gauche. C. R. Acad. Sci. Paris, Sér. A–B 289 (1979), no. 3, 151.
33.Bak, Anthony; Rehmann, Ulf; The congruence subgroup and metaplectic problems for SLn≥2 of division algebras. J. Algebra 78 (1982), no. 2, 475547.
34.Bak, Anthony; Rehmann, Ulf; K 2-analogs of Hasse's norm theorems. Comment.Math. Helv. 59 (1984), no. 1, 111.
35.Bak, Anthony; Scharlau, Winfried; Grothendieck and Witt groups of orders and finite groups Invent.Math. 23 (1974), 207240.
36.Bak, Anthony; Stepanov, Alexei, Dimension theory and nonstable K-theory for net groups. Rend. Sem. Mat. Univ. Padova 106 (2001), 207253.
37.Bak, Anthony, Tang, Guoping, Stability for hermitian K 1 J. Pure Appl. Algebra 150 (2000), no. 2, 109121.
38.Bak, Anthony; Tang, Guoping, Solution to the presentation problem for powers of the augmentation ideal of torsion free and torsion abelian groups. Adv. Math. 189 (2004), no. 1, 137.
39.Bak, Anthony; Vavilov, NikolaiNormality for elementary subgroup functors. Math. Proc. Cambridge Philos. Soc. 118 (1995), no. 1, 3547.
40.Bak, Anthony; Vavilov, Nikolai; Presenting powers of augmentation ideal and Pfister forms. K-Theory 20 (2000), no. 4, 299309.
41.Bak, Anthony; Vavilov, Nikolai, Structure of hyperbolic unitary groups. I. Elementary subgroups. Algebra Colloq. 7 (2000), no. 2, 159196
42.Bak, Anthony; Vavilov, Nikolai, Cubic form parameters, preprint.
43.Bak, Anthony; Hazrat, Roozbeh; Vavilov, Nikolai, Structure of hyperbolic unitary groups. II. Normal subgroups, preprint.
44.Bak, Anthony; Hazrat, Roozbeh; Vavilov, Nikolai, Localisation-completion strikes again: Relative K 1 is nilpotent by abelian, J. Pure Appl. Algebra 213, (2009) 10751085.
45.Bass, Hyman, K-theory and stable algebra, Publ. Math. Inst Hautes Etudes Sci. 22 (1964), 560.
46.Bass, Hyman, Algebraic K-theory, Benjamin, New York, 1968.
47.Bass, HymanUnitary algebraic K-theory. Lecture Notes Math. 343 (1973), 57265.
48.Bass, Hyman; Pardon, William, Some hybrid asymplectic group phenomena. J. Algebra 53 (1978), no. 2, 327333.
49.Bass, Hyman, John, Milnor, the algebraist, Topological Methods in modern mathematics (Stony Brook – 1991) Publish or Perish, Houston, 1993, 4584.
50.Bass, Hyman, Personal reminiscences of birth of algebraic K-theory. K-theory 30 (2003), 203209
51.Bass, Hyman; Milnor, John; Serre, Jean-Pierre, Solution of the congruence subgroup problem for SLn (n ≥ 3) and Sp2n (n ≥ 2). Publ. Math. Inst. Hautes Etudes Sci. 33 (1967), 59137.
52.Basu, Rabeya; Rao, Ravi. A.; Khanna, Reema; On Quillen's local global principle. Commutative algebra and algebraic geometry, 1730, Contemp. Math. 390, Amer. Math. Soc., Providence, RI, 2005.
53.Bayer-Fluckiger, Eva, Principe de Hasse faible pour les systèmes des formes quadratiques. J. reine angew. Math. 378 (1987), 5359.
54.Borel, Armand; Serre, Jean-Pierre; Le théoreme de Riemann-Roch, Bull Soc. Math. France 86 (1958), 97136.
55.Borewicz, Zenon; Vavilov, Nikolai, The distribution of subgroups in the full linear group over a commutative ring, Proc. Steklov Institute Math 3 (1985), 2746.
56.Browder, William; Surgery on simply-connected manifolds. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 65, Springer-Verlag, New York-Heidelberg, 1972
57.Chevalley, Claude; Theory of Lie groups, vol. I, Princeton Univ. Press, Princeton N.J., 1946; 15th printing, 1999.
58.Chevalley, Claude; Sur certains groupes simples. Tôhoku Math. J. 7 (1955), 1466.
59.Chevalley, Claude; Classification des groupes algébriques semi-simples. Collected works, vol.3, Springer-Verlag, Berlin, 2005, 1276.
60.Costa, Douglas, Keller, Gordon; Radix redux: normal subgroups of symplectic groups. J. reine angew. Math. 427 (1992), 51105.
61.Dennis, Keith, Stability for K 2. Lect. Notes. Math. 353 (1973), 8594.
62.Dickson, Leonard, Linear groups: with an exposition of the Galois field theory. Dover publications, N.Y., 1958.
63.Dickson, Leonard Eugene, Theory of linear groups in an arbitrary field. Trans. Amer. Math. Soc. 2 (1901), no. 4, 363394.
64.Dieudonné, Jean, Sur les groupes classiques. 3ème ed., Hermann. Paris, 1973.
65.Dieudonné, Jean, On the automorphism of the classical groups. Mem. Amer. Math. Soc. (1951), no. 2, 1122.
66.Dieudonné, Jean, La géometrie des groupes classiques. 3ème ed., Springer Verlag, Berlin, 1971.
67.Dieudonné, Jean, Panorama des mathématiques pures. Le Choix bourbachique. Gauthier-Villars, Paris, 1977.
68.Estes, David; Ohm, Jack, Stable range in commutative rings. J. Algebra, 7 (1967), no. 3, 343362.
69.Fröhlich, Albrecht, Hermitian and quadratic forms over rings with involution. Quart. J. Math. Oxford Ser.2, 20 (1969), 297317.
70.Gerasimov, Viktor, The group of units of a free product of rings, Math. U.S.S.R. Sbornik 62 (1989), no. 14163.
71.Golubchik, Igor, On the general linear group over an associative ring. Uspehi Mat. Nauk. 28 (1973), no. 3, 179180.
72.Golubchik, Igor, On the normal subgroups of orthogonal group over an associative ring with involution. Uspehi Mat. Nauk. 30 (1975), no. 6, 165.
73.Golubchik, Igor, The normal subgroups of linear and unitary groups over rings. Ph. D. Thesis, Moscow State Univ. (1981) 1117 (in Russian).
74.Golubchik, Igor, On the normal subgroups of the linear and unitary groups over associative rings. Spaces over algebras and some problems in the theory of nets. Ufa, 1985, 122142 (in Russian).
75.Golubchik, Igor, Mikhalev Alexander, Elementary subgroup of a unitary group over a PI-ring. Vestnik Mosk. Univ., ser.1, Mat., Mekh. (1985), no. 1, 3036.
76.Golubchik, Igor, Mikhalev, Alexander, On the elementary group over a PI-ring. Studies in Algebra, Tbilisi (1985), 2024. (in Russian).
77.Habdank, Günter, A classification of subgroups of Λ-quadratic groups normalized by relative elementary subgroups. Adv. Math. 110 (1995), no. 2, p.191233.
78.Hahn, Alexander; O'Meara, O. T., The classical groups and K-theory, Springer, Berlin 1989.
79.Hazrat, Roozbeh, Dimension theory and nonstable K 1 of quadratic modules, K-theory 27, (2002) 293328.
80.Hazrat, Roozbeh; Vavilov, Nikolai, K 1 of Chevalley groups are nilpotent. J. Pure Appl. Algebra 179, (2003) 99116.
81.Jordan, Camille, Traité des substitutions et des équations algébriques. Reprint of the 1870 original. Editions Jacques Gabay, Sceaux, 1989
82.van der Kallen, Wilberd, Injective stability for K 2. Lect. Notes Math. 551 (1976), 77156.
83.van der Kallen, Wilberd, Another presentation for Steinberg groups, Indag. Math. 39, no. 4 (1977), 304312.
84.van der Kallen, Wilberd, The K 2 of rings with many units, Ann. Sci. Ecole Norm. Sup. (4) 10 (1977), no. 4, 473515.
85.van der Kallen, Wilberd, A group structure on certain orbit sets of unimodular rows, J. Algebra 82 (1983), 363397.
86.van der Kallen, Wilberd, A module structure on certain orbit sets of unimodular rows, J. Pure Appl. Algebra 57 (1989), 281316.
87.van der Kallen, Wilberd; Magurn, Bruce; Vaserstein, Leonid; Absolute stable rank and Witt cancellation for non-commutative rings. Invent. Math. 91 (1988) 525542.
88.Kato, KazuyaSymmetric bilinear forms, quadratic forms and Milnor K-theory in characteristic two. Invent. Math. 66 (1982), no. 3, 493510.
89.Karoubi, Max; Periodicity theorems and conjectures in hermitian K-theory, an appendix to the current paper, J. K-theory 4 (2009), 6775.
90.Kervaire, Michel; Milnor, John, Groups of homotopy spheres. I. Ann. of Math. (2) 77 1963 504537.
91.Kervaire, Michel; Multiplicateurs de Schur et K-theory. Essays on Topology and Related Topics, Mém. dédiés à G. de Rham. Springer Verlag: Berlin et al. 1970, 212225.
92.Klein, Igor, Mikhalev, Alexander; The orthogonal Steinberg group over a ring with involution Algebra and Logic 9 (1970), 88103.
93.Klein, Igor, Mikhalev, Alexander; The unitary Steinberg group over a ring with involution Algebra and Logic 9 (1970), 305312.
94.Knus, Max-Albert, Quadratic and hermitian forms over rings. Springer Verlag, Berlin. 1991.
95.Knus, Max-Albert; Merkuriev, Alexander; Rost, Marcus; Tignol, Jean-Pierre. The Book of involutions. Maer. Math. Soc. Colloq. Publ. 44, 1998.
96.Kolster, Manfred; On injective stability for K 2. Algebraic K-theory, Part I (Oberwolfach, 1980), 128168, Lecture Notes in Math. 966, Springer, Berlin-New York, 1982.
97.Laitinen, Erkki; Morimoto, Masaharu; Pawalowski, Krzysztof; Deleting-inserting theorem for smooth actions of finite nonsolvable groups on spheres. Comment. Math. Helv. 70 (1995), no. 1, 1038.
98.Lam, Tsit-Yuen, The algebraic theory of quadratic forms. Benhamin. Reading, 1973.
99.Li, Fuan, The structure of symplectic group over arbitrary commutative rings. Acta Math. Sinica, New Series 3 (1987), no. 3, 247255.
100.Li, Fuan, The structure of orthogonal groups over arbitrary commutative rings. Chinese Ann. Math. 10B (1989), no. 3, 341350.
101.Li, Shangzhi, A new type of classical groups over skew-fields of characteristic 2. J.Algebra 138 (1991), no. 2, 399419.
102.Milnor, John, Algebraic K-theory and quadratic forms. Invent. Math. 9 (1970), 318344.
103.Milnor, John, Introduction to algebraic K-theory, Princeton Univ. Press, Princeton, N. J., 1971.
104.Milnor, John, On manifolds homeomorphic to the 7-sphere. Ann. of Math. (2) 64 (1956), 399405.
105.Milnor, John; Husemoller, Dale; Symmetric bilinear forms. Springer Verlag, Heidelberg, 1973.
106.Moore, Calvin; Group extensions of p-adic and adelic linear groups. Publ.Math. Inst. Hautes Etudes Sci. 35 (1968), 157222.
107.Morimoto, Masaharu, Bak groups and equivariant surgery. K-theory 2 (1989), no. 4, 465483.
108.Morimoto, Masaharu, Bak groups and equivariant surgery. II. K-theory 3 (1990), no. 6, 505521
109.Mundkur, Arun, Dimension theory and nonstable K 1. Algebr. Represent. Theory 5 (2002), no. 1, 355.
110.Novikov, Sergey, Homotopically equivalent smooth manifolds. I. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964) 365474
111.O'Meara, Timothy, Introduction to quadratic forms. Reprint of the 1973 ed. Springer Verlag, Berlin. 2000
112.Pender, William; Classical groups over division rings of charactristic two. Bull Austral. Math. Soc. 7 (1972), 191226. Correction, ibid 319.
113.Petrie, Ted, One fixed point actions on spheres. I, II. Adv. in Math. 46 (1982), no. 1, 314, 1570.
114.Petrov, Viktor, Overgroups of unitary groups. K-theory 29 (2003), no. 3, 147174.
115.Petrov, Viktor, Odd unitary groups. J. Math. Sci. 130 (2003), no. 3, 47524766.
116.Petrov, Viktor, Overgroups of classical groups. Ph.D. Thesis, Saint-Petersburg State Univ. (2005), 1129 (in Russian).
117.Platonov, Vladimir; Rapinchuk, Andrei, Algebraic groups and number theory, Academic Press.
118.Plotkin, Eugene, Stability theorems for K-functors for Chevalley groups, Proc. Conf. Nonassociative Algebras and Related Topics (Hiroshima — 1990), World Sci. London et al., 1991, 203217.
119.Plotkin, Eugene, Surjective stabilization for K 1-functor for some exceptional Chevalley groups. J. Soviet Math. 64, 1993, p.751767.
120.Plotkin, Eugene, On the stability of the K 1-functor for Chevalley groups of type E 7, J. Algebra 210 (1998), 6785.
121.Plotkin, Eugene; Stein, Michael; Vavilov, Nikolai, Stability of K-functors modeled on Chevalley groups, revisited. Unpublished manuscript (2001), 121.
122.Raghunathan, M. S., The congruence subgroup problem. Proc. Indian Acad. Sci. Math. Sci. 114 (2004), no. 4, 299308.
123.Rehmann, Ulf, Zentrale Erweiterungen der speziellen linearen Gruppe eines Schiefkörpers. J. reine angew. Math. 301 (1978), 77104.
124.Rehmann, Ulf, A survey of the congruence subgroup problem. Algebraic K-theory, Part I (Oberwolfach, 1980), 197207, Lecture Notes in Math. 966, Springer, Berlin-New York, 1982.
125.Sah, Chih Han; Symmetric bilinear forms and quadratic forms. J. Algebra 20 (1972), 144160.
126.Serre, Jean-Pierre; Modules projectifs et espaces fibrés a fibre vectorielle. Séminaire P. Dubreil, M.-L. Dubreil-Jacotin et C. Pisot, 1957/58, Fasc.2, Exposé 23, Paris, 1958
127.Scharlau, Winfried, Quadratic and Hermitian forms. Springer Verlag, Berlin. 1985.
128.Scharlau, Winfried, On the history of the algebraic theory of quadratic forms. Quadratic forms and their applications (Dublin, 1999), 229259, Contemp. Math. 272, Amer. Math. Soc., Providence, RI, 2000.
129.Sharpe, Richard, On the structure of the unitary Steinberg groups. Ann. Math. 96 (1972), 444479.
130.Smale, Stephen, Generalized Poincaré's conjecture in dimensions greater than four. Ann. of Math. 74 (1961), no. 2, 391406.
131.Stafford, J. Tobias, Absolute stable rank and quadratic forms over noncommutative rings. K-theory 4 (1990), 121130.
132.Stein, Michael; Relativizing functors on rings and algebraic K-theory, J. Algebra 19 (1971), 140152.
133.Stein, Michael; Stability theorems for K 1, K 2 and related functors modeled on Chevalley groups. Japan. J. Math. 4 (1978), no. 1, 77108.
134.Steinberg, Robert; Générateurs, rélations et revêtements des groupes algébriques. Colloque Théorie des Groupes Algébriques (Bruxelles—1962), 113127.
135.Steinberg, Robert; Lectures on Chevalley groups Yale University, 1967.
136.Stepanov, Alexei; Non-standard subgroups between E(n,R) and GL(n,A). Algebra Colloq. 11 (2004), no. 3, 321334.
137.Stepanov, Alexei, Subgroups of a Chevalley group between two different rings, Preprint.
138.Stepanov, Alexei; Vavilov, Nikolai; Decomposition of transvections: a theme with variations. K-theory 19 (2000), 109153.
139.Stepanov, Alexei; Vavilov, Nikolai; You, Hong Localisation-complition in the description of intermediate subgroups (2007), 143.
140.Suslin, Andrei, On the structure of the general linear group over polynomial rings, Soviet Math. Izv. 41 (1977), no. 2, 503516.
141.Suslin, Andrei, Stability in algebraic K-theory. Lect. Notes. Math., 996 (1980), 304333.
142.Suslin, Andrei, Algebraic K-theory and norm residue homomorphisms, J. Sov. Math. 30 (1985), 25562611.
143.Suslin, Andrei, Kopeiko, Viacheslav, Quadratic modules and orthogonal groups over polynomial rings. J. Sov. Math. 20 (1982), no. 6, 26652691.
144.Suslin, Andrei; Tulenbaev, Marat, Stabilization theorem for Milnor's K 2-functor. J. Sov. Math. 17 (1981), 18041819.
145.Tang, Guoping, Hermitian groups and K-theory, K-theory 13 (1998), no. 3, 209267.
146.Tang, Guoping, Presenting powers of augmentation ideals in elementary p-groups. K-theory 13 (2001) no. 1, 3139.
147.Tang, Guoping, Hermitian forms over local rings, Algebra Colloq. 8 (2001), no. 1, 110.
148.Tits, Jacques; Buildings of spherical type and finite BN-pairs. Lecture Notes Math. 386 (1973).
149.Tits, Jacques; Formes quadratiques, groupes orthogonaux et algébres de Clifford. Invent.Math. 5 (1968), 1941.
150.Tulenbaev, Marat, Schur, multiplier of the group of elementary matrices of finite order, J. Sov. Math. 17 (1981), no. 4, 20622067.
151.Vaserstein, Leonid, On the stabilization of the general linear group over a ring. Math USSR Sbornik 8 (1969), 383400.
152.Vaserstein, Leonid, K 1-theory and the congruence subgroup problem, Mat. Zametki 5 (1969), 233244.
153.Vaserstein, Leonid, Stabilization of unitary and orthogonal groups over a ring with involution. Math USSR Sbornik 10 (1970), 307326.
154.Vaserstein, Leonid, Stabilization for Milnor's K 2-functor. (Russian) Uspehi Mat. Nauk 30 (1975), no. 1 (181), 224.
155.Vaserstein, Leonid, On the normal subgroups of GLn over a ring, Lecture Notes Math. 854 (1981), 456465.
156.Vaserstein, Leonid; The subnormal structure of general linear groups over rings, Math. Proc. Camb. Phil. Soc, 108 (1990), 219229.
157.Vaserstein, Leonid; Normal subgroups of orthogonal groups over commutative rings. Amer. J. Math. 110 (1988), no. 5, 955973.
158.Vaserstein, Leonid; Normal subgroups of symplectic groups over rings. K-theory 2 (1989), no. 5, 647673.
159.Vaserstein, Leonid; You, Hong; Normal subgroups of classical groups over rings. J. Pure Appl. Algebra 105 (1995), 93106.
160.Vavilov, Nikolai; The structure of split classical groups over a commutative ring. Soviet Math. Dokl. 37 (1988), no. 2, 550553.
161.Vavilov, Nikolai; Structure of Chevalley groups over commutative rings. Proc. Conf. Non-Associative Algebras and Related Topics (Hiroshima – 1990). World Sci., London et al., 1991, 219335.
162.Vavilov, Nikolai; Intermediate subgroups in Chevalley groups. Proc. Conf. Groups of Lie type and their Geometries (Como – 1993). Cambridge Univ. Press, 1995, 233280.
163.Vavilov, Nikolai; Subnormal structure of general linear group, Math. Proc. Camb. Phil. Soc. 107 (1990), 103196.
164.Vavilov, Nikolai; A third look at weight diagrams. Rend. Sem. Mat. Univ. Padova. 104 (2000) no. 1, 201250.
165.Vavilov, Nikolai; An A 3-proof of the main structure theorems for Chevalley groups of types E 6 and E 7, Int. J. Algebra. Comput. 17 (2007), no. 56, 12831298.
166.Vavilov, Nikolai; Gavrilovich, Mikhail, A 2-proof of the structure theorems for Chevalley groups of types E 6 and E 7, St. Petersburg Math. J. 16 (2005), 649672.
167.Vavilov, Nikolai; Gavrilovich, Mikhail; Nikolenko, Sergei, Structure of Chevalley groups: the Proof from the Book, J. Math. Sci. 330 (2006), 3676.
168.Vavilov, Nikolai; Nikolenko, Sergei, A 2-proof of the structure theorems for Chevalley groups of types F4, St. Petersburg Math. J. 20 (2008), no. 3.
169.Vavilov, Nikolai; Petrov, Viktor, On overgroups of Ep(2l,R), St. Petersburg Math. J. 15 (2004), 515543.
170.Vavilov, Nikolai; Petrov, Viktor, On overgroups of EO(n,R, St. Petersburg Math. J. 19 (2007), no. 2, 1051.
171.Voevodsky, Vladimir, Motivic cohomology with ℤ/2-coefficients. Publ. IHES. 98 (2003), 59104.
172.van der Waerden, Bartels, Gruppen der linearen Transformationen. Chelsea, New York, 1948.
173.Wall, Charles Terence Clehh; On the axiomatic foundation of the theory of Hermitian forms. Proc. Cambridge. Phil. Soc. 67 (1970), 243250.
174.Wall, Charles Terence Clehh; Surgery of non-simply-connected manifolds. Ann. of Math. (2) 84 (1966), 217276.
175.Wall, Charles Terence Clehh; Surgery on compact manifolds. Second edition. Edited and with a foreword by Ranicki, A. A.. Mathematical Surveys and Monographs 69. American Mathematical Society, Providence, RI, 1999
176.Weil, Andre, Classical groups and algebras with involution, J. Indian Math. Soc. 24 (1961), 589623.
177.Weyl, Hermann, The classical groups. Their invariants and representations. 15th printing, Princeton Univ. Press, Princeton. 1998
178.Weyl, Hermann, Review of Dieudonné “Sur les groupes classiques”", MR0024439.
179.Wilson, John, The normal and subnormal structure of general linear groups, Proc. Camb. Phil. Soc. 71 (1972), 163177.
180.Witt, Ernst, Theorie der quadratischen Formen in beliebigen Körpern. J. reine angew. Math. 176 (1937), 3144.
181.Zhang, Zuhong, Stable sandwich classification theorem for classical-like groups, Math. Proc. Cambridge Phil. Soc. 143 (2007), no. 3, 607619.



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