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Integral group rings: Groups of units and classical K-theory

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Abstract

This survey contains results obtained in this area from the second half of the sixties to the present time. In the group of units of the group ring, normal periodic subgroups, elements of finite order, free subgroups, congruence subgroups, questions of conjugacy of finite subgroups, and matrix representations are considered. Moreover, questions connected with the calculation of the groups K0, K1 for group rings are discussed with a description of the structure of projective modules, groups of invertible matrices, etc.

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Literature cited

  1. V. A. Artamonov, “Nonfree protective modules over group rings of solvable groups,” Mat. Sb.,116, No. 2, 232–244 (1981).

    Google Scholar 

  2. V. A. Artamonov, “Projective modules over group rings of nilpotent groups,” in: Algebra (Collection of Papers Decicated to the 90th Birthday of O. Yu. Schmidt) [in Russian], Moscow (1982), pp, 7–23.

  3. V. A. Artamonov, “Structure of projective groups in products of manifolds,” Tr. Sem. I. G. Petrovsk. MGU, No. 8, 58–74 (1982).

    Google Scholar 

  4. H. Bass, Algebraic K-Theory [Russian translation], Mir, Moscow (1973).

    Google Scholar 

  5. S. D. Berman, “Properties of integral group rings,” Dokl. Akad. Nauk SSSR,91, No. 1, 7–9 (1953).

    Google Scholar 

  6. S. D. Berman, “A necessary condition for isomorphism in integer group rings,” Dokl. Akad. Nauk Ukr. SSR, No. 5, 313–316 (1953).

    Google Scholar 

  7. S. D. Berman, “The equation xm=1 in an integral group ring,” Ukr. Mat. Zh.,7, No. 3, 253–261 (1955).

    Google Scholar 

  8. S. D. Berman, “Representations of finite groups,” Itogi Nauka Tekhn., Ser. Algebra 1964, 83–122 (1966).

    Google Scholar 

  9. S. D. Berman and A. R. Rossa, “Integral group rings,” in: Third Scientific Conference of Young Mathematicians of the Ukraine [in Russian], Kiev (1966), p. 75.

  10. S. D. Berman and A. R. Rossa, “Integral group rings of finite and periodic groups,” in: Algebra and Mathematical Logic [in Russian], Izd. Kiev. Univ., Kiev (1966), pp. 44–53.

    Google Scholar 

  11. A. A. Bovdi, “Periodic normal subgroups of the multiplicative group of a group ring,” Sib. Mat. Zh.,9, No. 3, 495–498 (1968).

    Google Scholar 

  12. A. A. Bovdi, “Periodic normal subgroups of the multiplicative group of a group ring. II,” Sib. Mat. Zh.,11, No. 3, 492–511 (1970).

    Google Scholar 

  13. A. A. Bovdi, “Isomorphism of integral group rings,” in: Materials of the 29th Scientific Conference of the Professorial-Instructional Staff of Uzhgor. Univ., Sect. Mathematical Sciences [in Russian], Uzhgor. Univ., Uzhgorod (1975), pp. 104–109. Dep. in VINITI March 10, (1976), No. 705-76 Dep.

    Google Scholar 

  14. A. A. Bovdi, “Structure of an integral group ring with trivial elements of finite order,” Sib. Mat. Zh.,21, No. 4, 28–37 (1980).

    Google Scholar 

  15. A. A. Bovdi, “Periodic normal subgroups of the multiplicative group of an integral group ring,” in: Abstracts of Reports to the 16th National Algebraic Conference [in Russian], Izd. Leningr. Univ., Leningrad (1981), p. 21.

    Google Scholar 

  16. A. A. Bovdi, “Structure of group bases of a group ring,” Mat. Zametki,32, No. 4, 459–468 (1982).

    Google Scholar 

  17. A. A. Bovdi, “Unitariness of the multiplicative group of an integral group ring,” Mat. Sb.,119, No. 3, 387–400 (1982).

    Google Scholar 

  18. A. A. Bovdi, “Unitary subgroup and congruence-subgroup of the multiplicative group of an integral group ring,” Dokl. Akad. Nauk SSSR,284, No. 5, 1041–1044 (1985).

    Google Scholar 

  19. A. A. Bovdi, “Unitary subgorup of the multiplicative group of the integral group ring of a cyclic group,” Mat. Zametki,41, No. 4, 469–474 (1987).

    Google Scholar 

  20. A. A. Bovdi, “Multiplicative group of an integral group ring,” Dep. in UkrNIINTI, Sept. 24, 1987, No. 2712-Uk 87.

  21. A. A. Bovdi, “Structure of the multiplicative group of an integral group ring,” Dokl. Akad. Nauk SSSR,301, No. 6, 1295–1297 (1988).

    Google Scholar 

  22. A. A. Bovdi, Group Rings [in Russian], UMK VO, Kiev (1988).

    Google Scholar 

  23. A. A. Bovdi and M. A. Dokuchaev, “Conjugacy of group bases of an integral group ring,” Mat. Issled. (Kishinev), No. 85, 32–42 (1985).

    Google Scholar 

  24. N. Bourbaki, Elements of Mathematics. Commutative Algebra [Russian translation], Mir, Moscow (1971).

    Google Scholar 

  25. A. E. Zalesskii, “A conjecture of Kaplansky,” Dokl. Akad. Nauk SSSR,203, No. 4, 749–751 (1972).

    Google Scholar 

  26. A. E. Zalesskii and A. V. Mikhailev, “Group rings,” Itogi Nauki Tekhn., Ser. Sovrem. Probl. Matem.,2, 5–118 (1973).

    Google Scholar 

  27. M. A. Dokuchaev, “Multiplicative groups of group rings of groups of order 8,” Dep. UkrNIINTI, Dec. 29, 1987, No. 3316-Uk 87.

  28. M. A. Dokuchaev, “Generalized trace of an element of finite order of the multiplicative group of an integral group ring,” Dep. UkrNIINTI, October 25, 1988, No. 2718-Uk 88.

  29. E. M. Zhmud' and G. Ch. Kurennoi, “Finite group of units of an integral group ring,” Vestn. Khar'kovsk. Univ. Ser. Mekhan.-Mat.,26, 20–26 (1967).

    Google Scholar 

  30. E. Sh. Kerer, “Projective ideals of a crossed group algebra of a free Abelian gorup of rank two,” Dep. UkrNIINTI, Jan. 25, 1984, No. 101 Uk-D84.

  31. A. I. Kostrikin and I. A. Chubarov, “Representations of finite groups,” Itogi Nauki Tekhn., Ser. Algebra, Topologiya, Geometriya,23, 119–195 (1985).

    Google Scholar 

  32. C. Curtis and I. Reiner, Theory of Representations of Finite Groups and Associative Algebras [Russian translation], Nauka, Moscow (1969).

    Google Scholar 

  33. J. Milnor, Introduction to Algebraic K-Theory [Russian translation], Mir, Moscow (1974).

    Google Scholar 

  34. S. P. Novikov, “Algebraic structure and properties of Hermitian analogs of K-theory over rings with involution from the point of view of Hamiltonian formalism. Some applications to differential topology and the theory of characteristic classes. II,” Izv. Akad. Nauk SSSR, Ser. Mat.,34, No. 3, 475–500 (1970).

    Google Scholar 

  35. Z. F. Patai, “Center of the multiplicative group of an integral group ring,” in: Materials of the Second Conference of Young Scientists. Notes of the Scientific Center of the Academy of Sciences of the UkrSSR, Math. Sciences Section [in Russian], Uzhgor. Univ., Uzhgorod (1975), pp. 37–41.

    Google Scholar 

  36. V. P. Platonov, “The Tanaka-Artin problem and reduced K-theory,” Izv. Akad. Nauk SSSR, Ser. Mat.,40, No. 2, 227–261 (1976).

    Google Scholar 

  37. G. G. Rakiashvili, “Structure of projective modules over crossed group rings,” Soobshch. Akad. Nauk GruzSSR,101, No. 3, 549–552 (1981).

    Google Scholar 

  38. A. V. Roiter, “Integral representations belonging to one family,” Izv. Akad. Nauk SSSR,30, 1315–1324 (1966).

    Google Scholar 

  39. A. I. Saksonov, “Group rings of finite groups,” Publ. Math.,18, Nos. 1–4, 187–209 (1971).

    Google Scholar 

  40. A. A. Stolin, “The group K0 for the integral group ring of a cyclic group of order p2,” Dep. in VINITI, May 16, 1980, No. 1887-80 Dep.

  41. A. A. Stolin, “Poincaré group of the integral group ring of a primary cyclic group,” Dep. in UkrNIINTI, Jan. 27, 1987, No. 506-Uk 87.

  42. A. A. Suslin, “Structure of the special linear group over polynomial rings,” Izv. Akad. Nauk SSSR, Ser. Mat.,41, No. 2, 235–252 (1977).

    Google Scholar 

  43. A. A. Suslin, “Algebraic K-theory,” Itogi Nauki Tekhn., Ser. Algebra, Topologiya, Geometriya,20, 71–152 (1982).

    Google Scholar 

  44. I. I. Khripta, “Nilpotence of the multiplicative group of a group ring,” Latv. Mat. Ezhegodnik,13, 119–127 (1973).

    Google Scholar 

  45. T. Akasaki, “A note on nonfinitely generated projective ZII-modules,” Proc. Am. Math. Soc.,86, No. 3 (1982).

  46. P. J. Allen and C. Hobby, “A characterization of units in Z[A4],” J. Algebra,66, No. 2, 534–543 (1980).

    Google Scholar 

  47. P. J. Allen and C. Hobby, “A note on the unit group of ZS3,” Proc. Amer. Math. Soc.,99, No. 1, 9–14 (1987).

    Google Scholar 

  48. P. J. Allen and C. Hobby, “Units in integral group rings of some metacyclic groups,” Can. Math. Bull.,30, No. 2, 231–240 (1987).

    Google Scholar 

  49. P. J. Allen and C. Hobby, “A characterization of units in ZS4,” Commun. Algebra,16, No. 7, 1479–1505 (1988).

    Google Scholar 

  50. R. C. Alperin, R. K. Dennis, and M. R. Stein, “SK1 of finite Abelian groups. I,” Invent. Math.,82, No. 1, 1–18 (1985).

    Google Scholar 

  51. R. C. Alperin, R. K. Dennis, R. Oliver, and M. R. Stein, “SK1 of finite Abelian groups. II,” Invent. Math.,87, No. 2, 253–302 (1987).

    Google Scholar 

  52. V. A. Artamonov, “Nonfree projectives in products of group varieties,” in: Univ. Algebra and Appl. Stefan Banach Int. Math. Cent. Semest., Feb. 15–June 9, (1978), Warsaw (1982), pp. 7–13.

  53. R. G. Ayoub and C. Ayoub, “On the group ring of a finite Abelian group,” Bull. Austral. Math. Soc.,1, No. 2, 245–261 (1969).

    Google Scholar 

  54. H. Bass, “The Dirichlet unit theorem, induced characters, and Whitehead groups of finite groups,” Topology,4, No. 4, 391–410 (1966).

    Google Scholar 

  55. H. Bass, “Euler characteristics and characters of discrete groups,” Invent. Math.,35, 155–196 (1976).

    Google Scholar 

  56. P. H. Berridge and M. J. Dunwoody, “Non-free projective modules for torsion-free groups,” J. London Math. Soc.,19, No. 3, 433–436 (1979).

    Google Scholar 

  57. A. K. Bhandari, “Some remarks on the unit groups of integral group rings,” Arch. Math.,44, No. 4, 319–322 (1985).

    Google Scholar 

  58. A. K. Bhandari and I. S. Luthar, “Conjugacy classes of torsion units of the integral group ring of DP,” Commun. Algebra,11, No. 14, 1607–1627 (1983).

    Google Scholar 

  59. A. K. Bhandari and I. S. Luthar, “Torsion units of integral group rings of metacyclic groups,” J. Number Theory,17, No. 2, 270–283 (1983).

    Google Scholar 

  60. A. K. Bhandari and I. S. Luthar, “Certain conjugacy classes of units in integral group rings of metacyclic groups,” J. Number Theory,18, No. 2, 215–228 (1984).

    Google Scholar 

  61. K. A. Brown, T. H. Lenagan, and J. T. Stafford, “K-theory and stable structure of some Noetherian group rings,” Proc. London Math. Soc.,42, No. 2, 193–230 (1981).

    Google Scholar 

  62. B. Bürgisser, “On the projective class group of arithmetic groups,” Math. Z.,184, No. 3, 339–357 (1983).

    Google Scholar 

  63. Ph. Cassou-Noguès, “Classes d'idéaux d'algébre d'une groupe Abelean,” C. R. Acad. Sci. Paris,276, No. 14, A973-A975 (1973).

    Google Scholar 

  64. G. H. Cliff, “Ranks of projective modules of group rings,” Commun. Algebra,13, No. 5, 1115–1130 (1985).

    Google Scholar 

  65. G. H. Cliff and S. K. Sehgal, “On the trace of an idempotent in a group ring,” Proc. Amer. Math. Soc.,62, No. 1, 11–14 (1977).

    Google Scholar 

  66. G. H. Cliff and S. K. Sehgal, “Groups which are normal in the unit groups of their group rings,” Arch. Math.,33, No. 6, 529–537 (1980).

    Google Scholar 

  67. G. H. Cliff, S. K. Sehgal, and A. R. Weiss, “Units of integral group rings of met-Abelian groups,” J. Algebra,73, No. 1, 167–185 (1981).

    Google Scholar 

  68. G. H. Cliff and A. R. Weiss, “Moody's induction theorem,” Ill. J. Math.,32, No. 3, 489–500 (1988).

    Google Scholar 

  69. C. W. Curtis and I. Reiner, Methods of Representation Theory with Applications to Finite Groups and Orders. Vol. 1, John Wiley and Sons, New York (1981).

    Google Scholar 

  70. C. W. Curtis and I. Reiner, Methods of Representation Theory with Applications to Finite Groups and Orders. Vol. 2, John Wiley and Sons, New York (1987).

    Google Scholar 

  71. R. K. Dennis, “Units of group rings,” J. Algebra,43, No. 2, 655–664 (1976).

    Google Scholar 

  72. R. K. Dennis, “The structure of the unit group of group rings,” in: Ring Theory, Proc. 2nd Okla. Conf., 1975, New York-Basel (1977), pp. 103–130.

  73. R. K. Dennis, “NK1 of finite groups,” Proc. Amer. Math. Soc.,100, No. 2, 229–232 (1987).

    Google Scholar 

  74. W. Dicks, “Groups, trees and projective modules,” Lect. Notes Math.,790 (1980).

  75. M. J. Dunwoody, “The homotopy type of a two-dimensional complex,” Bull. London Math. Soc.,8, No. 3, 282–285 (1976).

    Google Scholar 

  76. S. Endo and Y. Horonaka, “Finite groups with trivial class groups,” J. Math. Soc. Japan,31, No. 1, 161–174 (1979).

    Google Scholar 

  77. D. R. Farkas, “Group rings: an annotated questionnaire,” Commun. Algebra,8, No. 6, 585–602 (1980).

    Google Scholar 

  78. D. R. Farkas and R. L. Snider, “K0 and Noetherian group rings,” J. Algebra,42, No. 1, 192–198 (1976).

    Google Scholar 

  79. F. T. Farrell and W. C. Hsiang, “The Whitehead group of poly-(finite or cyclic) groups,” J. London Math. Soc.,24, No. 2, 308–324 (1981).

    Google Scholar 

  80. T. Furukawa, “Note on groups with isomorphic group algebras,” Math. J. Okayama Univ.,24, No. 1, 1–6 (1982).

    Google Scholar 

  81. T. Furukawa, “The group of normalized units of a group ring,” Osaka J. Math.,23, No. 1, 217–221 (1986).

    Google Scholar 

  82. S. Galovich, I. Reiner, and S. Ullom, “Class groups for integral representations of metacyclic groups,” Mathematika,19, No. 1, 105–111 (1972).

    Google Scholar 

  83. J. Z. Goncalves, “Free subgroups of units in group rings,” Can. Math. Bull.,27, No. 3, 309–312 (1984).

    Google Scholar 

  84. J. Z. Goncalves, “Normal and subnormal subgroups in the group of units of group rings,” Bull. Austral. Math. Soc.,31, No. 3, 355–363 (1985).

    Google Scholar 

  85. J. Z. Goncalves, “Group rings with solvable unit groups,” Commun. Algebra,14, No. 1, 1–19 (1986).

    Google Scholar 

  86. J. Z. Goncalves, “Free subgroups and the residual nilpotence of the group of units of modular and p-adic group rings,” Can. Math. Bull.,29, No. 3, 321–328 (1986).

    Google Scholar 

  87. J. Z. Goncalves, J. Ritter, and S. Sehgal, “Subnormal groups in U(ZG),” Proc. Amer. Math. Soc.,103, No. 2, 375–382 (1988).

    Google Scholar 

  88. N. Gupta, “Free group rings,” Contemp. Math.,66, 1–129 (1987).

    Google Scholar 

  89. W. H. Gustafson and K. W. Roggenkamp, “A Mayer-Vietoris sequence for Picard groups with applications to integral group rings of dihedral and quaternion groups,” Ill. J. Math.,32, No. 3, 375–406 (1988).

    Google Scholar 

  90. B. Hartley and P. F. Pickel, “Free subgroups in the unit groups of integral group rings,” Can. J. Math.,32, No. 6, 1342–1352 (1980).

    Google Scholar 

  91. K. Hoechsmann and J. Ritter, “Logarithms and units in p-adic Abelian group rings,” Arch. Math.,49, No. 1, 23–28 (1987).

    Google Scholar 

  92. K. Hoechsmann and S. K. Sehgal, “Units in regular Abelian p-group rings,” J. Number Theory,30, No. 3, 375–381 (1988).

    Google Scholar 

  93. K. Hoechsmann and S. K. Sehgal, “Units in regular elementary Abelian group rings,” Arch. Math.,47, No. 5, 413–417 (1986).

    Google Scholar 

  94. I. Hughes and K. R. Pearson, “The group of units of the integral group ring ZS3,” Can. Math. Bull.,15, No. 4, 529–534 (1972).

    Google Scholar 

  95. D. A. Jackson, “The group of units of the integral group rings of finite met-Abelian and finite nilpotent groups,” Quart. J. Math.,20, No. 79, 319–331 (1969).

    Google Scholar 

  96. S. Jajodia and B. A. Magurn, “Surjective stability of units and simple homotopy type,” J. Pure Appl. Algebra,18, No. 1, 45–58 (1980).

    Google Scholar 

  97. A. V. Jategaonkar, “Localization in Noetherian rings,” Lect. Notes Math. London Math. Soc.,12, No. 98 (1986).

  98. E. Jespers and P. F. Smith, “Integral group rings of torsion-free polycyclic-by-finite groups are maximal orders,” Commun. Algebra,13, No. 3, 669–680 (1985).

    Google Scholar 

  99. G. Karpilovsky, “On the isomorphism problem for integral group rings,” J. Algebra,59, No. 1, 1–4 (1979).

    Google Scholar 

  100. G. Karpilovsky, Commutative Group Algebras, M. Dekker Inc., New York (1978).

    Google Scholar 

  101. I. Kaplansky, “Problems in the theory of rings revisites,” Amer. Math. Monthly,77, No. 5, 445–454 (1970).

    Google Scholar 

  102. S. Kleinert, “Einheiten in Z[D2m],” J. Number Theory,13, No. 14, 541–561 (1981).

    Google Scholar 

  103. E. Kleinert, “Units in Z[QP],” J. Number Theory,26, No. 2, 227–236 (1987).

    Google Scholar 

  104. M. E. Keating, “Whitehead groups of dihedral 2-groups,” Lect. Notes Math.,966, 122–127 (1982).

    Google Scholar 

  105. Kropholder, Linnell, and Moody, “Applications of a new K-theoretic theorem to soluble group rings,” Proc Amer. Math. Soc.,104, 675–684 (1988).

    Google Scholar 

  106. J. Lewin, “Projective modules over group algebras of torsion-free groups,” Mich. Math. J.,29, No. 1, 59–64 (1982).

    Google Scholar 

  107. P. A. Linnell, “Nonfree projective modules for integral group rings,” Bull. London Math. Soc.,14, No. 2, 124–126 (1982).

    Google Scholar 

  108. B. A. Magurn, “Whitehead groups of some hyperelementary groups,” J. London Math. Soc.,21, No. 1, 176–188 (1980).

    Google Scholar 

  109. B. A. Magurn, “Uses of units in Whitehead groups,” Lect. Notes Math.,882, 261–268 (1981).

    Google Scholar 

  110. B. A. Magurn, R. Oliver, and L. Vaserstein, “Units in Whitehead groups of finite groups,” Prepr. Ser. Mat. Inst. Aarhus Univ., 1981–1982, No. 27.

  111. Z. Marciniak, J. Ritter, S. K. Sehgal, and A. Weiss, “Torsion units in integral group rings of some met-Abelian groups. II,” J. Number Theory,25, No. 3, 340–352 (1987).

    Google Scholar 

  112. W. May, “Group algebras over finitely generated rings,” J. Algebra,39, No. 2, 483–511 (1976).

    Google Scholar 

  113. W. May, “Unit groups and isomorphism theorems for commutative group algebras,” in: Group and Semigroup Rings. Proc. Int. Conf. Johannesburg, 7–13 July, 1985, Amsterdam (1986), pp. 163–178.

  114. J. C. McConnell, “The K-theory of filtered rings and skew Laurent extensions,” Lect. Notes Math.,1146, 288–298 (1985).

    Google Scholar 

  115. J. C. McConnell and J. C. Robson, Noncommutative Noetherian Rings, John Wiley and Sons, Chichester (1987).

    Google Scholar 

  116. T. Mitsuda, “On the torsion units of integral dihedral group rings,” Commun. Algebra,14, No. 9, 1707–1728 (1986).

    Google Scholar 

  117. J. A. Moody, “Induction theorems for infinite groups,” Bull. Amer. Math. Soc.,17, No. 1, 113–116 (1987).

    Google Scholar 

  118. I. Musson and A. Weiss, “Integral group rings with residually nilpotent unit groups,” Arch. Math.,38, No. 6, 514–530 (1982).

    Google Scholar 

  119. I. Nagasaki, “Homotopy representation groups and Swan subgroups,” Osaka J. Math.,24, No. 2, 253–261 (1987).

    Google Scholar 

  120. R. Oliver, “SK1 for finite group rings. I,” Invent. Math.,57, No. 2, 183–204 (1980).

    Google Scholar 

  121. R. Oliver, “SK1 for finite group rings. II,” Prepr. Ser. Math. Inst. Aarhus Univ., 1979/1980,47, No. 25, 195–231.

    Google Scholar 

  122. R. Oliver, “SK1 for finite group rings. III,” Lect. Notes Math.,854, 299–337 (1981).

    Google Scholar 

  123. R. Oliver, “SK1 for finite group rings. IV,” Proc. London Math. Soc.,46, No. 1, 1–37 (1983).

    Google Scholar 

  124. R. Oliver, “Class groups of cyclic p-groups,” Mathematika,30, No. 1, 26–57 (1983).

    Google Scholar 

  125. R. Oliver, “Projective class groups of integral group rings: a survey,” Lect. Notes Math.,1142, 211–232 (1985).

    Google Scholar 

  126. R. Oliver, “Central units in p-adic group rings,” Prepr. Ser. Mat. Inst. Aarhus Univ., 1986–1987, No. 1.

  127. R. Oliver, “SK1 of finite group rings. V,” Comment. Math. Helv.,62, No. 3, 465–509 (1987).

    Google Scholar 

  128. R. Oliver, “Whitehead groups of finite groups,” London Math. Soc. Lecture Notes Ser., No. 132 (1988).

  129. I. B. S. Pasi and S. K. Sehgal, “Integral group rings,” J. Algebra,23, No. 2, 343–349 (1972).

    Google Scholar 

  130. D. C. Passman, The Algebraic Structure of Group Rings, John Wiley and Sons, New York (1977).

    Google Scholar 

  131. D. C. Passman, “Group rings, crossed products and galois theory,” Reg. Conf. Ser. Math., No. 64, Amer. Math. Soc. (1986).

  132. D. C. Passman, “Infinite crossed products,” Math. Dept., Univ. of Wisconsin (1989).

  133. D. C. Passman and P. F. Smith, “Units in integral group rings,” J. Algebra,69, No. 1, 213–239 (1981).

    Google Scholar 

  134. G. L. Peterson, “On the automorphism group of an integral group ring,” Arch. Math.,28, No. 6, 577–583 (1977).

    Google Scholar 

  135. M. C. Polcino, “The group of units of the integral group ring ZD4,” Bol. Soc. Brasil. Mat.,4, No. 2, 85–92 (1973).

    Google Scholar 

  136. M. C. Polcino, “Integral group rings with nilpotent unit groups,” Can. J. Math.,28, No. 5, 954–960 (1976).

    Google Scholar 

  137. M. C. Polcino, “Group rings whose units form an FC-group,” Arch. Math.,30, No. 4, 380–384 (1978).

    Google Scholar 

  138. M. C. Polcino, “Group rings whose torsion units form a subgroup,” Proc. Amer. Math. Soc.,81, No. 2, 172–174 (1981).

    Google Scholar 

  139. M. C. Polcino, “Group rings whose torsion units form a subgroup. II,” Commun. Algebra,9, No. 7, 699–712 (1981).

    Google Scholar 

  140. M. C. Polcino, “Units of group rings: a short survey,” London Math. Soc. Lect. Note Ser., No. 71, 281–297 (1982).

    Google Scholar 

  141. M. C. Polcino, “Torsion units in group rings and a conjecture of H. J. Zassenhaus,” in: Group and Semigroup Rings. Proc. Int. Conf., Johannesburg, 7–13 July, 1985, Amsterdam (1986), pp. 179–192.

  142. M. C. Polcino, J. Ritter, and S. K. Sehgal, “On conjecture of Zassenhaus on torsion units in integral group rings. II,” Proc. Amer. Math. Soc.,97, No. 2, 201–206 (1986).

    Google Scholar 

  143. M. C. Polcino and S. K. Sehgal, “Torsion units in integral group rings of metacyclic groups,” J. Number Theory,19, No. 1, 103–114 (1984).

    Google Scholar 

  144. I. Reiner, “Projective class groups of symmetric and alternating groups,” Linear Multilinear Algebra,3, Nos. 1–2, 147–153 (1975).

    Google Scholar 

  145. I. Reiner, “Class groups and Picard groups of group rings and orders,” Reg. Conf. Ser. Math.,26, CBMS, Amer. Math. Soc., 1–43 (1976).

  146. I. Reiner and K. W. Roggenkamp, “Integral representations and presentation of finite groups,” Lect. Notes Math.,744, 148–273 (1979).

    Google Scholar 

  147. J. Ritter and S. K. Sehgal, “Integral group rings of some p-groups,” Can. J. Math.,34, No. 1, 233–246 (1982).

    Google Scholar 

  148. J. Ritter and S. K. Sehgal, “On a conjecture of Zassenhaus on torsion units in integral group rings,” Math. Ann.,264, No. 2, 257–270 (1983).

    Google Scholar 

  149. J. Ritter and S. K. Sehgal, “Certain normal subgroups of units in group rings,” J. Reine Angew. Math.,381, 214–220 (1987).

    Google Scholar 

  150. K. W. Roggenkamp, “Structure of integral group rings,” Lect. Notes Math.,867, 421–440 (1981).

    Google Scholar 

  151. K. W. Roggenkamp, “Units in integral met-Abelian group rings. I. Jackson's unit theorem revisited,” Quart. J. Math.,32, No. 126, 209–224 (1981).

    Google Scholar 

  152. K. W. Roggenkamp and L. L. Scott, “Units in met-Abelian group rings: non-splitting examples for normalized units,” J. Pure Appl. Algebra,27, 299–314 (1983).

    Google Scholar 

  153. K. W. Roggenkamp and L. L. Scott, “Units in group rings: splittings and the isomorphism problem,” J. Algebra,96, No. 2, 397–417 (1985).

    Google Scholar 

  154. K. W. Ropgenkamp and L. L. Scott, “The isomorphism problem for integral group rings of finite nilpotent groups,” London Math. Soc. Lect. Note Ser., No. 121, 291–299 (1986).

    Google Scholar 

  155. K. W. Roggenkamp and L. L. Scott, “Isomorphisms of p-adic group rings,” Ann. Math.,126, No. 3, 593–647 (1987).

    Google Scholar 

  156. F. Röhl, “On the isomorphism problem for integral group-rings of circle-groups,” Math. Z.,180, No. 3, 419–422 (1982).

    Google Scholar 

  157. R. Sandling, “Note on the integral group ring problem,” Math. Z.,124, No. 3, 255–258 (1972).

    Google Scholar 

  158. R. Sandling, “Graham Higman's thesis ‘Units in group rings,’” Lect. Notes Math.,882, 93–116 (1981).

    Google Scholar 

  159. R. Sandling, “The isomorphism problem for group rings: a survey,” Lect. Notes Math.,1142, 256–288 (1985).

    Google Scholar 

  160. L. Scott, “Recent progress on isomorphism problem,” in: Arcata Conf. Represent. Finite Groups: Proc. Summer Res. Inst., Arcata, Calif., July 7–25, 1986. Part 1, Providence, R. I. (1987), pp. 259–273.

  161. S. K. Sehgal, “Units in commutative integral group rings,” Math. J. Okayama Univ.,14, No. 2, 135–138 (1970).

    Google Scholar 

  162. S. K. Sehgal, “On class sums in p-adic group rings,” Can. J. Math.,23, No. 3, 541–543 (1971).

    Google Scholar 

  163. S. K. Sehgal, Topics in Group Rings, M. Dekker, New York-Basel (1978).

    Google Scholar 

  164. S. K. Sehgal, Sur. K. Sehgal, and H. J. Zassenhaus, “Isomorphism of integral group rings of Abelian by nilpotent class two groups,” Commun. Algebra,12, Nos. 19–20, 2401–2407 (1984).

    Google Scholar 

  165. S. K. Sehgal and Al. Weiss, “Torsion units in integral group rings of some met-Abelian groups,” J. Algebra,103, No. 2, 490–499 (1986).

    Google Scholar 

  166. S. K. Sehgal and H. J. Zassenhaus, “Group rings whose units form an FC-group,” Math. Z.,153, No. 1, 29–35 (1977).

    Google Scholar 

  167. S. K. Sehgal and H. J. Zassenhaus, “Integral group rings with nilpotent unit groups,” Communs. Algebra,5, No. 2, 101–111 (1977).

    Google Scholar 

  168. K. Sekiguchi, “On the units of integral group rings,” Tokyo J. Math.,3, No. 1, 149–162 (1980).

    Google Scholar 

  169. K. Sekiguchi, “On the automorphism group of the p-adic group ring of a metacyclic p-group. II,” J. Algebra,100, No. 1, 191–213 (1986).

    Google Scholar 

  170. L. W. Small (ed.), Noetherian Rings and Their Applications, Math. Surv. Mon., No. 24, Amer. Math. Soc., Providence (1987).

    Google Scholar 

  171. J. T. Stafford, “On the stable range of right Noetherian rings,” Bull. London Math. Soc.,13, No. 1, 39–41 (1981).

    Google Scholar 

  172. J. T. Stafford, “Stably free projective right ideals,” Compot. Math.,54, No. 1, 63–78 (1985).

    Google Scholar 

  173. M. R. Stein, “Whitehead groups of finite groups,” Bull. Amer. Math. Soc.,84, No. 2, 201–212 (1978).

    Google Scholar 

  174. A. Strojnowski, “A note on U. P. groups,” Commun. Algebra,8, No. 3, 231–234 (1980).

    Google Scholar 

  175. A. Strojnowski, “Idempotents and zero divisors in group rings,” Commun. Algebra,14, No. 7, 1171–1185 (1986).

    Google Scholar 

  176. R. G. Swan, “Periodic resolutions for finite groups,” Ann. Math.,72, No. 2, 267–291 (1960).

    Google Scholar 

  177. R. G. Swan, “The Grothendick ring of a finite group,” Topology,2, No. 2, 85–110 (1963).

    Google Scholar 

  178. R. G. Swan, “Projective modules over Laurent polynomial rings,” Trans. Amer. Math. Soc.,237, 111–120 (1978).

    Google Scholar 

  179. R. G. Swan, “Projective modules over binary polyhedral groups,” J. Reine Angew. Math., No. 342, 66–172 (1983).

    Google Scholar 

  180. R. G. Swan, “Torsion-free cancellation over orders,” Ill. J. Math.,32, No. 3, 329–360 (1988).

    Google Scholar 

  181. R. G. Swan and E. G. Evans, “K-theory of finite groups and orders,” Lect. Notes Math.,149 (1970).

  182. M. J. Taylor, “The locally free class groups of prime power order,” Ill. J. Algebra,50, No. 2, 463–487 (1978).

    Google Scholar 

  183. M. J. Taylor, “The locally free class group of the symmetric group,” J. Math.,23, No. 4, 687–702 (1979).

    Google Scholar 

  184. S. V. Ullom, “Nontrivial lower bounds for class groups of integral group rings,” Ill. J. Math.,20, No. 2, 361–371 (1976).

    Google Scholar 

  185. A. Weiss, “Rigidity of p-adic p-torsion,” Ann. Math.,127, No. 2, 317–332 (1988).

    Google Scholar 

  186. A. Williamson, “On the conjugacy classes in an integral group ring,” Can. Math. Bull.,21, No. 4, 491–496 (1978).

    Google Scholar 

  187. A. Yilmaz, “A characterization of units in ZS4,” Hecettere Bull. Natur. Sci. and Eng.,14–15, 41–52 (1985–1986).

    Google Scholar 

  188. H. J. Zassenhaus, “On the torsion units of finite group rings,” Estud. Mat. Lisboa, 119–120 (1974).

  189. J. Ziegenbalg, “Isomorphe Gruppenringe Lokal endlicher Gruppen,” J. Reine Angew. Math.,277, 82–88 (1975).

    Google Scholar 

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Authors
  1. V. A. Artamonov
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Translated from Itogi Nauki i Tekhniki, Seriya Algebra, Topologiya, Geometriya, Vol. 27, pp. 3–43, 1989.

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Artamonov, V.A., Bovdi, A.A. Integral group rings: Groups of units and classical K-theory. J Math Sci 57, 2931–2958 (1991). https://doi.org/10.1007/BF01099283

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