Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-26T13:52:07.733Z Has data issue: false hasContentIssue false
  • English
  • Français

On polycyclic groups with isomorphic finite quotients

Published online by Cambridge University Press:  24 October 2008

Fritz Grunewald  and
Daniel Segal
Fritz Grunewald
Affiliation:
Universität Bielefeld, W. Germany
Daniel Segal
Affiliation:
Universität Bielefeld, W. Germany
Get access Rights & Permissions [Opens in a new window]

Extract

Following P. F. Pickel (5) we write (G) for the set of isomorphism classes of finite quotients of a group G. One of the outstanding problems in the theory of polycyclic groups is to determine whether there can be infinitely many non-isomorphic polycyclic groups G with a given (G). We solve a special case of this problem with our first main result:

Theorem 1. Let G be an abelian-by-cyclic polycyclic group. Then the polycyclic-by-finite groups H withlie in only finitely many isomorphism classes.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 84 , Issue 2 , September 1978 , pp. 235 - 246
Copyright
Copyright © Cambridge Philosophical Society 1978

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)Borel, A.Some finiteness properties of adèle groups over number fields. Publ. Math. I.H.E.S. 16 (1963), 5–30.CrossRefGoogle Scholar
(2)Borel, A. and Serre, J.-P.Théorèmes de finitude en cohomologie galoisienne. Comment. Math. Helv. 39 (1964), 111–164.CrossRefGoogle Scholar
(3)Borevich, Z. I. and Shafarevich, I. R.Number theory (New York, London, Academic Press, 1966).Google Scholar
(4)Lang, S.Algebraic number theory (Reading (Mass.), Addison–Wesley, 1970).Google Scholar
(5)Pickel, P. F.Nilpotent-by-finite groups with isomorphic finite quotients. Trans. Amer. Math. Soc. 183 (1973), 313–325.CrossRefGoogle Scholar
(6)Pickel, P. F.Fitting subgroups and profinite completions of polycyclic groups. J. Algebra 42 (1976), 41–45.CrossRefGoogle Scholar
(7)Remeslennikov, V. N.Groups that are residually finite with respect to conjugaoy (in Russian). Sibirsk. Mat. Z. 12 (5) (1971), 1085–1099 (English translation, 12 (1971), 783–792).Google Scholar
(8)Weil, A. Adèles and algebraic groups. I.A.S., Princeton (1961).Google Scholar