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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
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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

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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

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