Geodesic
On primitive representations of soluble groups of finite rank
Sbornik. Mathematics, Tome 191 (2000) no. 11, pp. 1707-1748 Cet article a éte moissonné depuis la source Math-Net.Ru

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In the paper it is proved, in particular, that a group is polycyclic if and only if it is soluble of finite rank, satisfies the ascending chain condition for normal subgroups and admits a faithful irreducible primitive representation over a field of characteristic zero. Methods are developed that enable one to study induced representations of nilpotent and soluble groups of finite rank. 
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     title = {On primitive representations of soluble groups of finite rank},
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     url = {http://geodesic.mathdoc.fr/item/SM_2000_191_11_a5/}
}
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A. V. Tushev. On primitive representations of soluble groups of finite rank. Sbornik. Mathematics, Tome 191 (2000) no. 11, pp. 1707-1748. http://geodesic.mathdoc.fr/item/SM_2000_191_11_a5/

[1] Harper D. L., “Primitivity in representations of polycyclic groups”, Math. Proc. Cambridge Philos. Soc., 88 (1980), 15–31 | DOI | MR | Zbl

[2] Harper D. L., “Primitive irreducible representations of nilpotent groups”, Math. Proc. Cambridge Philos. Soc., 82 (1977), 241–247 | DOI | MR | Zbl

[3] Zalesskii A. E., “O neprivodimykh predstavleniyakh konechnoporozhdennykh nilpotentnykh grupp bez krucheniya”, Matem. zametki, 9:2 (1971), 199–210 | MR | Zbl

[4] Tushev A. V., “Spectra of conjugated ideals in group algebras of abelian groups of finite rank and control theorems”, Glasgow Math. J., 38 (1996), 309–320 | DOI | MR | Zbl

[5] Roseblade J. E., “Group rings of polycyclic groups”, J. Pure Appl. Algebra, 3 (1973), 307–328 | DOI | MR | Zbl

[6] Brookes C. J. B., “Modules over polycyclic groups”, Proc. London Math. Soc. (3), 57 (1988), 88–108 | DOI | MR | Zbl

[7] Brookes C. J. B., “Stabilisers in injective modules over nilpotent groups”, Proc. of Singapore Group Theory Conference (1987), de Gruyter, Berlin, 1989, 275–291 | MR

[8] Brookes C. J. B., Brown K. A., “Primitive group rings and Noetherian rings of quotients”, Trans. Amer. Math. Soc., 288 (1985), 605–623 | DOI | MR | Zbl

[9] Wehrfritz B. A. F., “Invariant maximal ideals in certain group algebras”, J. London Math. Soc. (2), 46 (1992), 101–110 | DOI | MR | Zbl

[10] Tushev A. V., “On modules over group rings of soluble groups of finite rank”, London Math. Soc. Lecture Note Ser., 261, 1999, 718–727 | MR | Zbl

[11] Charin V. S., “O gruppakh avtomorfizmov nilpotentnykh grupp”, Ukr. matem. zhurn., 3:6 (1954), 295–304 | MR

[12] Glushkov V. M., “O normalizatorakh polnykh podgrupp v polnoi gruppe”, Dokl. AN SSSR, 71:3 (1950), 421–424 | MR | Zbl

[13] Hall P., “Finiteness conditions for soluble groups”, Proc. London Math. Soc. (3), 4 (1954), 419–436 | DOI | MR | Zbl

[14] Wilson J. S., “Some properties of groups inherited by normal subgroups of finite index”, Math. Z., 144 (1970), 19–21 | DOI | MR

[15] Kropholler P. H., “On finitely generated soluble groups with large wreath product sections”, Proc. London Math. Soc. (3), 49 (1984), 155–169 | DOI | MR | Zbl

[16] Roseblade J. E., “Applications of the Artin–Rees lemma to group rings”, Simp. Math., 17, Academic Press, London, 1976, 471–478 | MR

[17] McConnell J. C., Robson J. C., Noncommutative Noetherian rings, Wiley, Chechester, 1987 | MR | Zbl

[18] Segal D., “On the residual simplicity of certain modules”, Proc. London Math. Soc. (3), 34 (1977), 327–353 | DOI | MR | Zbl

[19] Passman D. S., The algebraic structure of group rings, Wiley, New York, 1977 | MR | Zbl

[20] Wilson J. S., “Soluble products of minimax groups, and nearly surjective derivations”, J. Pure Appl. Algebra, 53 (1988), 297–331 | DOI | MR

[21] Brookes C. J. B., “Ideals in group rings of soluble groups of finite rank”, Math. Proc. Cambridge Philos. Soc., 97 (1985), 27–49 | DOI | MR | Zbl

[22] Tushev A. V., “Neterovy moduli nad abelevymi gruppami konechnogo svobodnogo ranga”, Ukr. matem. zhurn., 43 (1991), 1042–1048 | MR

[23] Burbaki N., Kommutativnaya algebra, Mir, M., 1971 | MR

[24] Tushev A. V., “O primitivnosti gruppovykh algebr nekotorykh klassov razreshimykh grupp”, Matem. sb., 186:3 (1995), 143–160 | MR | Zbl

[25] Tushev A. V., “Induced modules over group rings algebras of metabelian groups of finite rank”, Comm. Algebra, 27:12 (1999), 5921–5938 | DOI | MR | Zbl

[26] Brookes C. J. B., Brown K. A., “Injective modules, induction maps and endomorphism rings”, Proc. London Math. Soc. (3), 67 (1993), 127–158 | DOI | MR | Zbl

[27] Brown K. A., “The Nullstellensatz for certain group rings”, J. London Math. Soc. (2), 26 (1982), 425–434 | DOI | MR | Zbl

[28] McConnell J. C., “The Nullstellensatz and Jacobson properties for rings of differential operators”, J. London Math. Soc. (2), 26 (1982), 37–42 | DOI | MR | Zbl

[29] Hall P., “On the finiteness of certain soluble groups”, Proc. London Math. Soc. (3), 9 (1959), 595–622 | DOI | MR | Zbl