Hostname: page-component-77c89778f8-vpsfw Total loading time: 0 Render date: 2024-07-20T14:17:53.767Z Has data issue: false hasContentIssue false
  • English
  • Français

Representations of metabelian groups satisfying the minimal condition for normal subgroups

Published online by Cambridge University Press:  17 April 2009

Howard L. Silcock
Howard L. Silcock
Affiliation:
Department of Pure Mathematics, University of Adelaide, Adelaide, South Australia.
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.

A question of John S. Wilson concerning indecomposable representations of metabelian groups satisfying the minimal condition for normal subgroups is answered negatively, by means of an example. It is shown that such representations need not be irreducible, even when the group being represented is an extension of an elementary abelian p–group by a quasicyclic q–group of the type first described by V.S. Čarin, and the characteristic of the field is a prime distinct from both p and q. This implies that certain techniques used in the study of metabelian groups satisfying the minimal condition for normal subgroups are not available for the corresponding class of soluble groups of derived length 3.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 14 , Issue 2 , April 1976 , pp. 267 - 278
Copyright
Copyright © Australian Mathematical Society 1976

References

[1]Чарин, В.С. [Čarin, V.S.], “Замечание об условии минимальности для подгрупп” [A remark on the minimal condition for subgroups], Dokl. Akad. Nauk SSSR (N.S.) 66 (1949), 575–576.Google Scholar
[2]Hartley, B. and McDougall, D., “Injective modules and soluble groups satisfying the minimal condition for normal subgroups”, Bull. Austral. Math. Soc. 4 (1971), 113–135.CrossRefGoogle Scholar
[3]Heineken, H. and Wilson, J.S., “Locally soluble groups with min-n”, J. Austral. Math. Soc. 17 (1974), 305–318.Google Scholar
[4]Kovács, L.G. and Newman, M.F., “Direct complementation in groups with operators”, Arch. Math. (Basel) 13 (1962), 427–433.Google Scholar
[5]LeVeque, William Judson, Topics in number theory, Vol. I (Addison-Wesley, Reading, Massachusetts, 1956).Google Scholar
[6]McDougall, David, “Soluble groups with the minimum condition for normal subgroups”, Math. Z. 118 (1970), 157–167.CrossRefGoogle Scholar
[7]Neumann, B.H., “Twisted wreath products of groups”, Arch. Math. (Basel) 14 (1963), 1–6.CrossRefGoogle Scholar
[8]Robinson, Derek J.S., Finiteness conditions and generalized soluble groups, Part 1 (Ergebnisse der Mathematik und ihrer Grenzgebiete, 62. Springer-Verlag, Berlin, Heidelberg, New York, 1972).Google Scholar
[9]Silcock, Howard L., “Metanilpotent groups satisfying the minimal condition for normal subgroups”, Math. Z. 135 (1974), 165–173.CrossRefGoogle Scholar
[10]Silcock, Howard L., “On the construction of soluble groups satisfying the minimal condition for normal subgroups”, Bull. Austral. Math. Soc. 12 (1975), 231–257.Google Scholar
[11]Wilson, John S., “Some properties of groups inherited by normal subgroups of finite index”, Math. Z. 114 (1970), 19–21.Google Scholar
[12]Wilson, John S., “Locally soluble groups satisfying the minimal condition for normal subgroups”, Proc. Mini-Conference Group Theory, Australian National University, Canberra, 1975 (Lecture Notes in Mathematics, to appear).Google Scholar