Hostname: page-component-76fb5796d-dfsvx Total loading time: 0 Render date: 2024-04-26T02:29:35.272Z Has data issue: false hasContentIssue false
  • English
  • Français

Groups whose irreducible representations have finite degree

Published online by Cambridge University Press:  24 October 2008

B. A. F. Wehrfritz
B. A. F. Wehrfritz
Affiliation:
Queen Mary College, London
Get access Rights & Permissions [Opens in a new window]

Extract

Throughout this paper F denotes a (commutative) field. Let XF denote the class of all groups G such that every irreducible FC-module has finite dimension over F. In (1) P. Hall showed that if F is not locally finite and if G is polycyclic, then G∈XF if and only if G is abelian-by-finite. Also in (1), if F is locally finite he proved that every finitely generated nilpotent group is in XF and he conjectured that XF should contain every polycyclic group. This turned out to be very difficult, but a positive solution was eventually found by Roseblade, see (8). Meanwhile Levič in (4) had started a systematic investigation of the classes XF. Although his paper contains a number of errors, obscurities and omissions, it remains an interesting work, and it, or more accurately its recent translation, stimulated this present paper.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 90 , Issue 3 , November 1981 , pp. 411 - 421
Copyright
Copyright © Cambridge Philosophical Society 1981

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Hall, P.On the finiteness of certain soluble groups. Proc. London Math. Soc. (3), 9 (1959), 595–622.CrossRefGoogle Scholar
(2)Kaplansky, I.Commutative rings (Allyn and Bacon, Boston, Mass., 1970).Google Scholar
(3)Kuroš, A. G.The theory of groups, vol. 1 (Chelsea, New York, 1956).Google Scholar
(4)Levič, E. M.On torsion-free groups whose irreducible representations over some field are all finite-dimensional (Russian). Latvijas Valsts Univ. Zinātn. Raksti 151 (1971), 125–139.= Amer. Math. Soc. Transl. (2) 113 (1979), 225–234.Google Scholar
(5)Passman, D. S.Some isolated subsets of infinite solvable groups. Pacif. J. Math. 45 (1973), 221–228.CrossRefGoogle Scholar
(6)Passman, D. S.The algebraic structure of group rings (John Wiley & Sons, New York, 1977).Google Scholar
(7)Robinson, D. J. S.Finiteness conditions and generalized soluble groups (2 vols). (Springer, Berlin-Heidelberg-New York, 1972).Google Scholar
(8)Roseblade, J. E.Group rings of polycyclic groups. J. Pure Appi. Algebra 3 (1973), 307–328.CrossRefGoogle Scholar
(9)Wehrfritz, B. A. F.Infinite linear groups (Springer, Berlin-Heidelberg-New York, 1973).CrossRefGoogle Scholar
(10)Wehrfritz, B. A. F.Nilpotence in groups of semi-linear maps. II. J. London Math. Soc. (2), 16 (1977), 449–457.CrossRefGoogle Scholar
(11)Zalesskiiˇ, A. E.The Jacobson radical of the group algebra of a soluble group is locally nilpotent (Russian). Izv. Akad. Nauk SSSR, Ser. Mat. 38 (1974), 983–994.Google Scholar