Hostname: page-component-7dd5485656-bt4hw Total loading time: 0 Render date: 2025-10-31T00:17:17.681Z Has data issue: false hasContentIssue false
  • English
  • Français

On certain group ring problems

Published online by Cambridge University Press:  17 April 2009

G. Karpilovsky
G. Karpilovsky
Affiliation:
Department of Mathematics, La Trobe University, Bundoora, Victoria 3083, Australia.
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Recent developments on the isomorphism and other group ring problems are amply reviewed in Sehgal's book, Topics in group rings. The aim of this expository paper is to complement the content of Sehgal's book. Our main emphasis is the presentation of some results due to Saksonov which are published in Russian and do not seem well-known to the English reader. We also draw the reader's attention to some unpublished results of Higman.

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 21 , Issue 3 , June 1980 , pp. 329 - 350
Copyright
Copyright © Australian Mathematical Society 1980

References

[1]Bergman, George M. and Dicks, Warren, “On universal derivations”, J. Algebra 36 (1975), 193–211.Google Scholar
[2] С.Д. Берман [Berman, S.D.], “О необходимом Услови изоморФизма целочисленных групповых колец” [On a necessary condition for isomorphism of integral group rings”, Dopovĭdĭ. Akad. Nauk Ukraïn. RSR No. 5 (1953), 313–316.Google Scholar
[3] С.Д. Берман [Berman, S.D.], “О неноторых свойствах целочисленных групповых колец” [On certain properties of integral group rings], Dokl. Akad. Nauk SSSR (N.S.) 91 (1953), 7–9.Google Scholar
[4] С.Д. Берман [Berman, S.D.], “Об изоморФизмв центров групповых колец р-груии” [On the isomorphism of the centers of group rings of p-groups], Dokl. Akad. Nauk SSSR (N.S.) 91 (1953), 185–187.Google Scholar
[5] С.Д. Берман [Berman, S.D.], “Об уравнении x m = 1 в челочисленном групповом нольче” [On the equation xm = 1 in an integral group ring], Ukrain. Mat. Ž. 7 (1955), 253–261.Google Scholar
[6] С.Д. Берман [Berman, S.D.], “О некоторых своиствах групповых колец нед полем рациональных чисал” [On certain properties of group rings over the field of rational numbers], Užgorod. Gos. Univ. Naučn. Zap. Him. Fiz. Mat. 12 (1955), 88–110.Google Scholar
[7] С.Д. Берман [Berman, S.D.], “Групповые алгебры абелевых расширений конечных групп” [Group algebras of abelian extensions of finite groups], Dokl. Akad. Nauk SSSR (N.S.) 102 (1955), 431–434.Google Scholar
[8]Berman, S.D., “Group algebras of countable abelian p-groups”, Soviet Math. Dokl. 8 (1967), 871–873Google Scholar
[9] С.Д. Берман [Berman, S.D.], “Групповые алгебры счетных абелевых p-групп” [Group algebras of countable abelian p-groups], Publ. Math. Debrecen 14 (1967), 365–405.CrossRefGoogle Scholar
[10] С.Д. Берман Т.Ж. Моллов [Berman, S.D., Mollov, T.Ž.], “О групповых кольцах абелевых p-групп любой мощности” [The group rings of abelian p-groups of arbitrary power], Mat. Zametki 6 (1969), 381–392.Google Scholar
[11] С.Д. Берман, А.Р. Росса [Berman, S.D., Rossa, A.R.], “О целочисленных групповых кольцах конечных и перодичесних групп” [Integral group-rings of finite and periodic groups”, АлϨебра u мамемамuческая лоϨuка. АлϨебраuческuе uссле∂ованuя [Algebra and mathematical logic: Studies in algebra], 44–53 (Izdat. Kiev. University, Kiev, 1966).Google Scholar
[12]Bourbaki, N., Éléments de mathématique, Fasc. XXIII. Livre II : Algèbre. Chapitre 8: Modules et anneaux semisimples (Nouveau tirage de l'édition de 1958. Actualités Scientifiques et Industrielles, No. 1261. Hermann, Paris, 1973).Google Scholar
[13] А.А. Бовди [Bovdi, A.A.], “Периодическив нормалькые делители мультипликативной группы группоѕого кольца” [Periodic normal divisors of the multiplicative group of a group ring], Sibirsk. Mat. Ž. 9 (1968); 495–498.Google Scholar
[14] А.А. Бовди [Bovdi, A.A.], “Периодическив нормалькые делители мультипликативной группы группоѕого кольца. II” [Periodic normal subgroups of the multiplicative group of a group ring. II], Sibirsk Mat.Ž. 11 (1970), 492–511.Google Scholar
[15] А.А. Бовди [Bovdi, A.A.], Групповые кольца [Group rings] (Užgorod. Gosudarstv. Univ., Užgorod, 1974).Google Scholar
[16]Brauer, Richard, “Zur Darstellungstheorie der Gruppen endlicher Ordnung”, Math. Z. 63 (1955/1956), 406–444.Google Scholar
[17]Cohn, James A. and Livingstone, Donald, “On the structure of group algebras, I”, Canad. J. Math. 17 (1965), 583–593.CrossRefGoogle Scholar
[18]Curtis, Charles W., Reiner, Irving, Representation theory of finite groups (Pure and Applied Mathematics, 11. Interscience [John Wiley & Sons], New York, London, 1962).Google Scholar
[19]Dade, Everett C., “Deux groupes finis distincts ayant la m^ene algè bra de groupe sur tout corps”, Math. Z. 119 (1971), 345–348.CrossRefGoogle Scholar
[20]Fröhlich, A., “The Picard group of noncommutative rings, in particular or orders”, Trans. Amer. Math. Soc. 180 (1973), 1–45.Google Scholar
[21]Higman, Graham, “Units in group rings” (D. Phil, thesis, University of Oxford, Oxford, 1940).Google Scholar
[22]Higman, Graham, “The units of group–rings”, Proc. London Math. Soc. (2) 46 (1940), 231–248.Google Scholar
[23]Hughes, I. and Pearson, K.R., “The group of units of the integral group ring ZS 3”, Canad. Math. Bull. 15 (1972), 529–534.CrossRefGoogle Scholar
[24]Jackson, D.A., “The groups of units of the integral group rings of finite metabelian and finite nilpotent groups”, Quart. J. Math. Oxford (2) 20 (1969), 319–331.CrossRefGoogle Scholar
[25]Karpilovsky, G., “On the isomorphism problem for integral group rings”, J. Algebra 59 (1979), 1–45.Google Scholar
[26]Karpilovsky, G., “On group rings of finite metabelian groups”, J. Austral. Math. Soc. Ser. A 28 (1979), 378–384.Google Scholar
[27]Karpilovsky, G., “Finite groups with isomorphic group algebras”, Illinois J. Math. (to appear).Google Scholar
[28]Karpilovsky, G., “On some properties of group rings”,.J. Austral. Math. Soc. Ser. A (to appear).Google Scholar
[29] А.И. Лихтман [Lihtman, A. I.], “О групповых кольцах p-групп” [On group rings of p-groups], Izv. Akad. Nauk SSSR Ser. Mat. 27 (1963), 795–800.Google Scholar
[30]Miller, G.A., Blichfeldt, H.F., Dickson, L.E., Theory and applications of finite groups (J. Wiley & Sons, New York; Chapman and Hall, London; 1916. Reprinted and corrected: Stechert, New York, 1938. Republished: Dover, New York, 1961).Google Scholar
[31]Obayashi, Tadao, “Solvable groups with isomorphic group algebras”, J. Math. Soc. Japan 18 (1966), 394–397.Google Scholar
[32]Obayashi, Tadao, “Integral group rings of finite groups”, Osaka J. Math. 7 (1970), 253–266.Google Scholar
[33]Passman, D.S., “Isomorphic groups and group rings”, Pacific J. Math. 15 (1965), 561–583.Google Scholar
[34]Passman, Donald S., The algebraic structure of group rings (Interscience [John Wiley & Sons], New York, London, Sydney, 1977).Google Scholar
[35]Pearson, K.R., “On the units of a modular group ring”, Bull. Austral. Math. Soc. 7 (1972), 169–182.Google Scholar
[36]Pearson, K.R., “On the units of a modular group ring II”, Bull. Austral. Math. Soc. 8 (1973), 435–442.CrossRefGoogle Scholar
[37]Pearson, K.R. and Taylor, D.E., “Groups subnormal in the units of their modular group rings”, Proc. London Math. Soc. (3) 33 (1976), 313–328.CrossRefGoogle Scholar
[38]Peterson, Gary L., “Automorphisms of the integral group ring of Sn”, Proc. Amer. Math. Soc. 59 (1976), 14–18.Google Scholar
[39] С.С. Поляк [Poljak, S.S.], “Необходимое условие изоморФизма групполых колец над кольцом” [On a necessary condition for isomorphism of group rings over a ring], Dokl. Užgorod Cos. Univ. No. 3 (1960), 62.Google Scholar
[40]Roggenkamp, K.W., “Group rings of metabelian groups and extension categories”, Canad. J. Math. (to appear).Google Scholar
[41] А.И. Саконов [Saksonov, A.I.], “О цалочислвнном нольца харантаров ноначной группы” [The integral ring of characters of a finite group”, Vesai Akad. Navuk BSSR Ser. Fiz.-Mat. Navuk 1966, No. 3, 69–76.Google Scholar
[42] А.И. Саконов [Saksonov, A.I.], “О наноторых целочислаччых нольцах, ассоциированных с ноначной группой” [Certain integer-valued rings associated with a finite group”, Dokl. Akad. Nauk SSSR 171 (1966), 529–532.Google Scholar
[43] А.И. Саконов [Saksonov, A.I.], “О групповых нольцах нонечных p–группнад накоторыми областями целостности” [On group rings of finite p-groups over certain integral domains”, Dokl. Akad. Nauk SSSR 11 (1967), 204–207.Google Scholar
[44] А.И. Саконов [Saksonov, A.I.], “О групповых кольцах коначных групп I” [Group rings of finite groups I], Publ. Math. Debrecen 18 (1971), 187–209 (1972).CrossRefGoogle Scholar
[45]Sandling, Robert, “The modular group rings of p-groups” (PhD thesis, University of Chicago, Chicago, 1969).Google Scholar
[46]Sandling, Robert, “Subgroups dual to dimension subgroups”, Proc. Cambridge Philos. Soc. 71 (1972), 33–38.CrossRefGoogle Scholar
[47]Sandling, Robert, “Note on the integral group ring problem”, Math. Z. 124 (1972), 255–258.CrossRefGoogle Scholar
[48]Sandling, Robert, “Dimension subgroups over arbitrary coefficient rings”, J. Algebra 21 (1972), 250–265.CrossRefGoogle Scholar
[49]Sandling, Robert, “Group rings of circle and unit groups”, Math. Z. 140 (1974), 195–202.Google Scholar
[50]Sehgal, Sudarshan K., “Isomorphism of p-adic group rings”, J. Number Theory 2 (1970), 500–508.Google Scholar
[51]Sehgal, Sudarshan K., “On class sums in p-adic group rings”, Canad. J. Math. 23 (1971), 541–543.Google Scholar
[52]Sehgal, Sudarshan K., Topics in group rings (Monographs and Textbooks in Pure and Applied Mathematics, 50. Marcel Dekker, New York and Basel, 1978).Google Scholar
[53]Taylor, D.E., “Groups whose modular group rings have soluble unit groups”, Group theory, 112–117 (Proc. Miniconf. Theory of Groups, Canberra, 1975 Lecture Notes in Mathematics, 573. Springer-Verlag, Berlin, Heidelberg, New York, 1977).Google Scholar
[54]Weller, William R., “The units of the integral group ring ZD4” (PhD thesis, Pennsylvania State University, Pennsylvania, 1972).Google Scholar
[55]Whitcomb, Albert, “The group ring problem” (PhD thesis, University of Chicago, Chicago, 1968).Google Scholar
[56] Э.М. Жмудь, Г.Ч. Нуренной [Žmud, È.M., Kurennoĭ, G.Č.], “О ноначных группах единиц целочисленного группового кольца” [The finite groups of units of an integral group ring], Vestnik Har'kov. Gos. Univ. 1967, no. 26, 20–26.Google Scholar