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Primitivity in representations of polycyclic groups

Published online by Cambridge University Press:  24 October 2008

D. L. Harper
D. L. Harper
Affiliation:
Wycliffe Hall, Oxford
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Extract

In an earlier paper (5) we showed that a finitely generated nilpotent group which is not abelian-by-finite has a primitive irreducible representation of infinite dimension over any non-absolute field. Here we are concerned primarily with the converse question: Suppose that G is a polycyclic-by-finite group with such a representation, then what can be said about G?

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 88 , Issue 1 , July 1980 , pp. 15 - 31
Copyright
Copyright © Cambridge Philosophical Society 1980

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References

REFERENCES

(1)Atiyah, M. F. and Macdonald, I. G.Introduction to commutative algebra (Addison-Wesley, 1969).Google Scholar
(2)Bergman, G. M.The logarithmic limit set of an algebraic variety. Trans. Amer. Math. Soc. 157 (1971), 459–469.CrossRefGoogle Scholar
(3)Clifford, A. H.Representations induced in an invariant subgroup. Ann. of Math. 38 (1937), 533–550.CrossRefGoogle Scholar
(4)Hall, P.On the finiteness of certain soluble groups. Proc. London Math. Soc. (3) 9 (1959), 595–622.CrossRefGoogle Scholar
(5)Harper, D. L.Primitive irreducible representations of nilpotent groups. Math. Proc. Cambridge Philos. Soc. 82 (1977), 241–247.CrossRefGoogle Scholar
(6)Harper, D. L. Primitive irreducible representations of polycyclic groups. Ph.D. Thesis, University of Cambridge 1977.CrossRefGoogle Scholar
(7)Musson, I. M. Irreducible modules and their injeotive hulls over group rings. Ph.D. Thesis, University of Warwick 1979.Google Scholar
(8)Passman, D. S.The algebraic structure of group rings (Wiley-Interscience, 1977).Google Scholar
(9)Robinson, D. J. S.Hypercentral ideals, Noetherian modules and a theorem of Stroud. J. Algebra 32 (1974), 234–239.CrossRefGoogle Scholar
(10)Roseblade, J. E.Group rings of polycyclic groups. J. Pure Appl. Algebra 3 (1973), 307–328.CrossRefGoogle Scholar
(11)Roseblade, J. E.Prime ideals in group rings of polycyclic groups. Proc. London Math. Soc. (3) 36 (1978), 385–447.CrossRefGoogle Scholar
(12)Segal, D.Irreducible representations of finitely generated nilpotent groups. Math. Proc. Cambridge Philos. Soc. 81 (1977), 201–208.CrossRefGoogle Scholar
(13)Snider, R. L.Primitive ideals in group rings of polycyclic groups. Proc. Amer. Math. Soc. 57 (1976), 8–10.CrossRefGoogle Scholar
(14)Stroud, P. W. Topics in the theory of verbal subgroups. Ph.D. Thesis, University of Cambridge 1966.Google Scholar
(15)Suprunenko, D.Soluble and nilpotent linear groups. Translations Math. Monographs, vol. 9 (Amer. Math. Soc, 1963).Google Scholar