Projective representations of finite groups over number rings
Sbornik. Mathematics, Tome 11 (1970) no. 3, pp. 391-410 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We solve the problem of finding the number $n(R,G)$ of nondecomposable projective representations of a finite group $G$ over the ring $R$ of all integers of a finite extension $F$ of the field of rational $p$-adic numbers $Q$. Also we clear up the question as to when all indecomposable projective $R$-representations of a group $G$ are realized by left ideals of crossed group rings of the group $G$ and the ring $R$. We note that for ordinary $R$-representations of a group $G$ the problem of the finiteness of the number $n(R,G)$ was investigated by S. D. Berman, I. Reiner, A. Heller, H. Yacobinski and one of the authors of the present article. Bibliography: 30 titles.
@article{SM_1970_11_3_a6,
     author = {L. F. Barannik and P. M. Gudivok},
     title = {Projective representations of finite groups over number rings},
     journal = {Sbornik. Mathematics},
     pages = {391--410},
     year = {1970},
     volume = {11},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1970_11_3_a6/}
}
TY  - JOUR
AU  - L. F. Barannik
AU  - P. M. Gudivok
TI  - Projective representations of finite groups over number rings
JO  - Sbornik. Mathematics
PY  - 1970
SP  - 391
EP  - 410
VL  - 11
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/SM_1970_11_3_a6/
LA  - en
ID  - SM_1970_11_3_a6
ER  - 
%0 Journal Article
%A L. F. Barannik
%A P. M. Gudivok
%T Projective representations of finite groups over number rings
%J Sbornik. Mathematics
%D 1970
%P 391-410
%V 11
%N 3
%U http://geodesic.mathdoc.fr/item/SM_1970_11_3_a6/
%G en
%F SM_1970_11_3_a6
L. F. Barannik; P. M. Gudivok. Projective representations of finite groups over number rings. Sbornik. Mathematics, Tome 11 (1970) no. 3, pp. 391-410. http://geodesic.mathdoc.fr/item/SM_1970_11_3_a6/

[1] P. M. Gudivok, L. F. Barannik, “Pro proektivni zobrazheniya skinchennikh grup nad kiltsyami”, Dopovidi AN URSR, 1968, no. 4, 294–297 | MR | Zbl

[2] L. F. Barannik, P. M. Gudivok, “Pro nerozkladni proektivni zobrazheniya skinchennikh grup”, Dopovidi AN USSR, 1969, no. 5, 391–393 | MR | Zbl

[3] J. Schur, “Über die Darstellimg der endlicher Gruppen durch gebrochene lineare Substitutionen”, J. Reine Angew. Math., 127 (1904), 20–50 | Zbl

[4] J. Schur, “Untersuchungen über die Darstellimg der endlichen Gruppen durch gebrochene lineare Substitutional”, J. Reine Angew. Math., 132 (1907), 85–137 | Zbl

[5] K. Jamazaki, “On projective representations and ring extensions of finite groups”, J. Fac. Sci. Univ. Tokyo, 10:2 (1964), 147–195 | MR

[6] K. Jamazaki, “A note on projective representations of finite groups”, Scient. Papers Coll. Gen. Educ. Univ. Tokyo, 14:1 (1964), 27–36 | MR

[7] S. B. Conlon, “Twisted group algebras and their representations”, J. Austral. Math. Sop., 4 (1964), 152–173 | DOI | MR | Zbl

[8] D. G. Higman, “Indecomposable representations at characteristic $p$”, Duke Math. J., 21:2 (1954), 377–381 | DOI | MR | Zbl

[9] J. A. Green, “On the indecomposable representations of a finite group”, Math. Z., 70 (1959), 430–445 | DOI | MR | Zbl

[10] S. D. Berman, P. M. Gudivok, “O tselochislennykh predstavleniyakh konechnykh grupp”, DAN SSSR, 145:6 (1962), 1199–1201 | MR | Zbl

[11] A. Heller, I. Reiner, “On groups with finitely many indecomposable integral representations”, Bull. Amer. Math. Soc., 68:3 (1962), 210–212 | DOI | MR | Zbl

[12] P. M. Gudivok, “O predstavleniyakh konechnykh grupp nad kvadratichnymi koltsami”, DAN SSSR, 159:6 (1964), 1210–1213 | MR | Zbl

[13] P. M. Gudivok, “Predstavleniya konechnykh grupp nad chislovymi koltsami”, Izv. AN SSSR, seriya matem., 31 (1967), 799–834 | MR | Zbl

[14] H. Jacobinski, “Sur les ordres commutatifs avec un hombre fini de reseaux indecomposables”, Acta Math., 118 (1967), 1–31 | DOI | MR | Zbl

[15] A. Jones, “Groups with a finite number of indecomposable integral representations”, Michigan Math. J., 10:3 (1963), 257–261 | DOI | MR | Zbl

[16] Z. I. Borevich, D. K. Faddeev, “Teoriya gomologii v gruppakh. II”, Vestnik LGU, seriya matem., 7 (1959), 72–87 | Zbl

[17] H. Hasse, Zahlentheorie, Acad. Verlag, Berlin, 1963 | MR

[18] C. Chevalley, “L'aritmetique dans les algebras de matrices”, Actualités Sci. Ind., 1936, 323 | Zbl

[19] E. G. Straus, O. Taussky, “Remark on the preceding paper “Algebraic equations satisfied by roots of natural numbers””, Pacif. J. Math., 6:1 (1956), 97–98 | MR | Zbl

[20] Z. I. Borevich, D. K. Faddeev, “Predstavleniya poryadkov s tsiklicheskim indeksom”, Trudy matem. in-ta AN SSSR im. V. A. Steklova, LXXX, 1965, 51–65 | MR | Zbl

[21] Z. I. Borevich, D. K. Faddeev, “Zamechanie o poryadkakh s tsiklicheskim indeksom”, DAN SSSR, 164:4 (1965), 727–728 | Zbl

[22] H. Zassenhaus, “Neuer Beweis der Endlichkeit der Klassenzahl bei unimodular Aquivalenz endlicher ganzzahliger Substitutionsgruppen”, Hamb. Abh., 12 (1938), 276–288 | DOI | Zbl

[23] Z. I. Borevich, I. P. Shafarevich, Teoriya chisel, Nauka, Moskva, 1964

[24] Yu. A. Drozd, A. V. Roiter, “Kommutativnye koltsa s konechnym chislom tselochislennykh nerazlozhimykh predstavlenii”, Izv. AN SSSR, seriya matem., 31 (1967), 783–798 | MR | Zbl

[25] I. Reiner, H. Zassenhaus, “Equivalence of representations under extensions of local ground rings”, Illinois J. Math., 5:3 (1961), 409–441 | MR

[26] S. D. Berman, P. M. Gudivok, “Nerazlozhimye predstavleniya konechnykh grupp nad koltsom tselykh $p$-adicheskikh chisel”, Izv. AN SSSR, seriya matem., 28 (1964), 875–910 | MR | Zbl

[27] A. V. Roiter, “O predstavleniyakh tsiklicheskoi gruppy chetvertogo poryadka tselochislennymi matritsami”, Vestnik LGU seriya matem., 19 (1960), 65–74 | MR | Zbl

[28] D. K. Faddeev, “Ob odnoi teoreme teorii gomologii v gruppakh”, DAN SSSR, 92:4 (1953), 703–705 | MR | Zbl

[29] M. Kholl, Teoriya grupp, IL, Moskva, 1962

[30] M. Kneser, “Einige Bemerkungen über ganzzahliger Darstellungen endlicher Gruppen”, Arch. Math., XVII (1966), 377–379 | DOI | MR