Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-28T16:09:02.842Z Has data issue: false hasContentIssue false
  • English
  • Français

Effective aspects of profinite groups1

Published online by Cambridge University Press:  12 March 2014

Rick L. Smith
Rick L. Smith*
Affiliation:
Cornell University, Ithaca, New York 14853
Get access Rights & Permissions [Opens in a new window]

Extract

Profinite groups are Galois groups. The effective study of infinite Galois groups was initiated by Metakides and Nerode [8] and further developed by LaRoche [5]. In this paper we study profinite groups without considering Galois extensions of fields. The Artin method of representing a finite group as a Galois group has been generalized (effectively!) by Waterhouse [14] to profinite groups. Thus, there is no loss of relevance in our approach.

The fundamental notions of a co-r.e. profinite group, recursively profinite group, and the degree of a co-r.e. profinite group are defined in §1. In this section we prove that every co-r.e. profinite group can be effectively represented as an inverse limit of finite groups. The degree invariant is shown to behave very well with respect to open subgroups and quotients. The work done in this section is basic to the rest of the paper.

The commutator subgroup, the Frattini subgroup, the p-Sylow subgroups, and the center of a profinite group are essential in the study of profinite groups. It is only natural to ask if these subgroups are effective. The following question exemplifies our approach to this problem: Is the center a co-r.e. profinite group? Theorem 2 provides a general method for answering this type of question negatively. Examples 3,4 and 5 are all applications of this theorem.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 46 , Issue 4 , December 1981 , pp. 851 - 863
Copyright
Copyright © Association for Symbolic Logic 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.)

Footnotes

1

The results presented here are part of the author's Ph.D. dissertation written under Stephen G. Simpson. The author gratefully acknowledges the influence of his advisor on this work.

References

REFERENCES

[1]Feferman, S., Impredicativity of the existence of the largest divisible subgroup of an abelian p-group, Model theory and algebra: A memorial tribute to Abraham Robinson, Lecture Notes in Mathematics, vol. 498, Springer-Verlag, Berlin, 1975.CrossRefGoogle Scholar
[2]Fröhlich, A. and Sheperdson, J.C., Effective procedures in field theory, Proceedings of the Royal Society London, vol. 248 (1956), pp. 407–432.Google Scholar
[3]Huppert, B., Endliche Gruppen. I, Die Grundlehren der Mathematische Wissenschaften, Band 134, Springer, New York and Berlin, 1967.Google Scholar
[4]Kalantari, I. and Retzlaff, A., Maximal vector spaces under automorphisms of the lattice of recursively enumerable vector spaces, this Journal, vol. 42 (1977), pp. 481–491.Google Scholar
[5]LaRoche, P., Effective Galois theory, this Journal (to appear).Google Scholar
[6]Lin, C., Recursively presented abelian groups: effective p-group theory. I, this Journal (to appear).Google Scholar
[7]Magnus, W., Karrass, A. and Solitar, D., Combinatorial group theory: Presentations of groups in terms of generators and relations, second revised edition, Dover, New York, 1976.Google Scholar
[8]Metakides, G. and Nerode, A., Effective content of field theory, Annals of Mathematical Logic, vol. 17 (1979), pp. 289–320.CrossRefGoogle Scholar
[9]Rabin, M.O., Computable algebra, general theory and theory of computable fields, Transactions of the American Mathematical Society, vol. 95 (1960), pp. 341–360.Google Scholar
[10]Remmel, J.B., On r.e. and co-r.e. vector spaces with non-extendable bases, this Journal, vol. 45 (1980), pp. 20–34.Google Scholar
[11]Rogers, H., Theory of recursive functions and effective computability, McGraw-Hill, New York, 1968.Google Scholar
[12]Shatz, S., Profinite groups, arithmetic and geometry, Princeton University Press, Princeton, New Jersey, 1972.CrossRefGoogle Scholar
[13]Smith, R.L., The theory of profinite groups with effective presentations, Ph.D. Dissertation, Pennsylvania State University, University Park, 1979.Google Scholar
[14]Waterhouse, W., Profinite groups are Galois groups, Proceedings of the American Mathematical Society, vol. 42 (1974), pp. 639–640.Google Scholar