Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-27T19:29:07.675Z Has data issue: false hasContentIssue false
  • English
  • Français

Groups with no nontrivial linear representations

Published online by Cambridge University Press:  17 April 2009

A.J. Derrick
A.J. Derrick
Affiliation:
Department of MathematicsNational University of SingaporeLower Kent Ridge Rd Singapore 0511Republic of Singapore e-mail: matberic@nusunix.nus.sg
Rights & Permissions [Opens in a new window]

Abstract

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.

We study the class of groups having no nontrivial linear representations over certain fields. After showing the class to be closed under perfect extensions with locally soluble kernel, we expand considerably the number of acyclic groups known to be in the class, by application to both binate groups and the acyclic automorphism groups of de la Harpe and McDuff.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 50 , Issue 1 , August 1994 , pp. 1 - 11
Copyright
Copyright © Australian Mathematical Society 1994

References

[1]Alperin, R.C. and Berrick, A.J., ‘Linear representations of binate groups’, J. Pure Appl. Algebra (to appear).Google Scholar
[2]Baumslag, G., Dyer, E. and Heller, A., ‘The topology of discrete groups’, J. Pure Appl. Algebra 16 (1980), 1–47.Google Scholar
[3]Berrick, A.J., ‘Two functors from abelian groups to perfect groups’, J. Pure Appl Algebra 44 (1987), 35–43.Google Scholar
[4]Berrick, A.J., ‘Universal groups, binate groups and acyclicity’, in Group Theory, Proceedings of the Singapore Conference (Walter de Gruyter, Berlin, 1989), pp. 253–266.Google Scholar
[5]Berrick, A.J., ‘Remarks on the structure of acyclic groups’, Bull. London Math. Soc. 22 (1990), 227–232.CrossRefGoogle Scholar
[6]Berrick, A.J., ‘Torsion generators for all abelian groups’, J. Algebra 139 (1991), 190–194.Google Scholar
[7]Berrick, A.J. and Miller, C.F. III, ‘Strongly torsion generated groups’, Math. Proc. Cambridge Philos. Soc. III (1992), 219–229.Google Scholar
[8]Berrick, A.J. and Robinson, D.J.S., ‘Imperfect groups’, J. Pure Appl. Algebra 88 (1993), 3–22.CrossRefGoogle Scholar
[9]Berrick, A.J. and Varadarajan, K., ‘Binate towers of groups’, Arch. Math. 62 (1994), 97–111.CrossRefGoogle Scholar
[10]Beyl, F.R. and Tappe, J., Group extensions, representations, and the Schur multiplier, Lecture Notes in Math. 958 (Springer-Verlag, Berlin, Heidelberg, New York, 1982).Google Scholar
[11]Chen, Y., ‘On the representations of skew linear groups’, Bull. London Math. Soc. 21 (1989), 267–269.CrossRefGoogle Scholar
[12]Formanek, E. and Procesi, C., ‘The automorphism group of a free group is not linear’, J. Algebra 149 (1992), 494–499.Google Scholar
[13]Fuchs, L., Infinite Abelian groups II (Academic Press, New York, 1973).Google Scholar
[14]Hall, J.I. and Hartley, B., ‘A group theoretical characterization of simple, locally finite, finitary linear groups’, Arch. Math. 60 (1993), 108–114.CrossRefGoogle Scholar
[15]de la Harpe, P. and McDuff, D., ‘Acyclic groups of automorphisms’, Comment. Math. Heh. 58 (1983), 48–71.Google Scholar
[16]Heineken, H. and Wilson, J.S., ‘Locally soluble groups with min-n’, J. Austral. Math. Soc. 17 (1974), 305–318.CrossRefGoogle Scholar
[17]Heller, A., ‘On the homotopy theory of topogenic groups and groupoids’, Illinois J. Math. 24 (1980), 576–605.Google Scholar
[18]Higman, G., ‘A finitely generated infinite simple group’, J. London Math. Soc. 26 (1951), 61–64.CrossRefGoogle Scholar
[19]Kegel, O.H., ‘Über einfache, lokal endliche Gruppen’, Math.Z. 95 (1967), 169–195.CrossRefGoogle Scholar
[20]Kegel, O.H. and Wehrfritz, B.A.F., Locally finite groups, North-Holland Math. Library 3 (North-Holland, Amsterdam, 1973).Google Scholar
[21]Passman, D.S., Infinite group rings, Pure and Appl. Monographs 6 (Marcel Dekker, New York, 1971).Google Scholar
[22]Robinson, D.J.S., Finiteness conditions and generalized soluble groups I (Springer-Verlag, Berlin, Heidelberg, New York, 1972).Google Scholar
[23]Robinson, D.J.S., A course in the theory of groups, Graduate Texts in Mathematics 80 (Springer-Verlag, Berlin, Heidelberg, New York, 1982).Google Scholar
[24]Segal, G., ‘Classifying spaces related to foliations’, Topology 17 (1978), 367–382.Google Scholar
[25]Wehrfritz, B.A.F., Infinite linear groups (Springer-Verlag, Berlin, Heidelberg, New York, 1973).CrossRefGoogle Scholar
[26]Wilson, J.S., ‘Groups with every proper quotient finite’, Math. Proc. Cambridge Philos. Soc. 69 (1971), 373–390.Google Scholar
[27]Winter, D.J., ‘Representations of locally finite groups’, Bull. Amer. Math. Soc. 74 (1968), 145–148.CrossRefGoogle Scholar