Hostname: page-component-76fb5796d-vfjqv Total loading time: 0 Render date: 2024-04-28T11:05:45.400Z Has data issue: false hasContentIssue false
  • English
  • Français

Describing Groups

Published online by Cambridge University Press:  15 January 2014

André Nies
André Nies*
Affiliation:
Department of Computer Science, Auckland University, Auckland, New Zealand, E-mail: andre@cs.auckland.ac.nz
Get access Rights & Permissions [Opens in a new window]

Abstract

Two ways of describing a group are considered. 1. A group is finite-automaton presentable if its elements can be represented by strings over a finite alphabet, in such a way that the set of representing strings and the group operation can be recognized by finite automata. 2. An infinite f.g. group is quasi-finitely axiomatizable if there is a description consisting of a single first-order sentence, together with the information that the group is finitely generated. In the first part of the paper we survey examples of FA-presentable groups, but also discuss theorems restricting this class. In the second part, we give examples of quasi-finitely axiomatizable groups, consider the algebraic content of the notion, and compare it to the notion of a group which is a prime model. We also show that if a structure is bi-interpretable in parameters with the ring of integers, then it is prime and quasi-finitely axiomatizable.

Type
Articles
Information
Bulletin of Symbolic Logic , Volume 13 , Issue 3 , September 2007 , pp. 305 - 339
Copyright
Copyright © Association for Symbolic Logic 2007

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] Akiyama, S., Frougny, F., and Sakharovitch, J., Powers of rationals modulo 1 and rational base systems, preprint, 2005.Google Scholar
[2] Blumensath, A. and Grädel, E., Finite presentations of infinite structures: automata and interpretations, Theory of Computing Systems, vol. 37 (2004), pp. 641–674.Google Scholar
[3] Cannon, J. et al., Word processing in groups, Jones and Bartlett Publishers, Boston, MA, 1992.Google Scholar
[4] Fuchs, L., Infinite Abelian groups, vol. 2, Academic Press, 1973.Google Scholar
[5] Gromov, M., Groups of polynomial growth and expanding maps, Publications Mathématiques IHÉS, vol. 53 (1981), pp. 53–78.Google Scholar
[6] Hirshon, R., Some cancellation theorems with applications to nilpotent groups, Journal of Australian Mathematical Society (series A), vol. 23 (1977), pp. 147–165.CrossRefGoogle Scholar
[7] Hodges, W., Model Theory, Cambridge University Press, Cambridge, 1993.CrossRefGoogle Scholar
[8] Hodgson, B. R., Théories décidables par automate fini, Ph.D. thesis, University of Montreal, 1976.Google Scholar
[9] Hodgson, B. R. Théories décidables par automate fini, Annales des Sciences Mathématiques du Quebec, vol. 7 (1983), pp. 39–57.Google Scholar
[10] Kargapolov, M. and Merzljakov, J., Fundamentals of the theory of groups, Springer-Verlag, 1979.Google Scholar
[11] Kaye, R., Models of Peano Arithmetic, Oxford University Press, Oxford, 1991.CrossRefGoogle Scholar
[12] Khelif, A., Bi-interpretabilité et structures QFA: étude des groupes résolubles et des anneaux commutatifs, to appear.Google Scholar
[13] Khoussainov, B. and Nerode, A., Automatic presentations of structures, Logic and computational complexity (Leivant, D., editor), Lecture Notes in Computer Science, vol. 960, Springer-Verlag, 1995, pp. 367–392.CrossRefGoogle Scholar
[14] Khoussainov, B., Nies, A., Rubin, S., and Stephan, F., Automatic structures: richness and limitations, Proceedings of the 19th IEEE Symposium on Logic in Computer Science, Lecture Notes in Computer Science, Springer-Verlag, 2004, pp. 110–119.Google Scholar
[15] Lang, S., Algebra, Addison-Wesley, 1965.Google Scholar
[16] Mal'cev, A., On a correspondence between rings and groups, American Mathematical Society Translations, vol. 45 (1965), pp. 221–231.Google Scholar
[17] Morozov, A. and Nies, A., Finitely generated groups and first-order logic, Journal of the London Mathematical Society, vol. 71 (2005), no. 2, pp. 545–562.CrossRefGoogle Scholar
[18] Nies, A., Aspects of free groups, Journal of Algebra, vol. 263 (2003), pp. 119–125.Google Scholar
[19] Nies, A. Separating classes of groups by first-order formulas, International Journal of Algebra and Computation, vol. 13 (2003), pp. 287–302.CrossRefGoogle Scholar
[20] Nies, A. Comparing quasi-finitely axiomatizable groups and prime groups, Journal of Group Theory, to appear.Google Scholar
[21] Nies, A. and Semukhin, P., Finite automaton presentable abelian groups, Proceedings of LFCS 2007, to appear.Google Scholar
[22] Nies, A. and Thomas, R., Finite automaton presentable groups and rings, Journal of Algebra, to appear.Google Scholar
[23] Noskov, G. A., The elementary theory of a finitely generated almost solvable group, Izvestiya Akademiya Nauk SSSR Seriya Matematika, vol. 47 (1983), no. 3, pp. 498–517.Google Scholar
[24] Oger, F., Équivalence élémentaire entre groupes finis-par-abéliens de type fini, Commentarii Mathematici Helvetici, vol. 57 (1982), no. 3, pp. 469–480.Google Scholar
[25] Oger, F., Cancellation and elementary equivalence of finitely generated finite-by-nilpotent groups, Journal of the London Mathematical Society, vol. 30 (1991), pp. 293–299.Google Scholar
[26] Oger, F. Quasi-finitely axiomatizable groups and groups which are prime models, Journal of Group Theory, vol. 9 (2006), no. 1, pp. 107–116.Google Scholar
[27] Oger, F. and Sabbagh, G., Quasi-finitely axiomatizable nilpotent groups, Journal of Group Theory, vol. 9 (2006), no. 1, pp. 95–106.Google Scholar
[28] Oliver, G. P. and Thomas, R. M., Automatic presentations for finitely generated groups, STACS 2005 (Diekert, V. and Durand, B., editors), Lecture Notes in Computer Science, vol. 3404, Springer-Verlag, 2005, pp. 693–704.Google Scholar
[29] Robinson, D., A course in the theory of groups, Springer-Verlag, 1988.Google Scholar
[30] Rubin, S., Automatically presentable structures, to appear.Google Scholar
[31] Scanlon, T., Infinite finitely generated fields are bi-interpretable with N, to appear.Google Scholar
[32] Silver, J., Counting the number of equivalence classes of Borel and coanalytic equivalence relations, Annals of Pure and Applied Logic, vol. 18 (1980), pp. 1–28.Google Scholar
[33] Sipser, M., Introduction to the theory of computation, PWS Publishing Company, 1997.Google Scholar
[34] Thomas, W., Automata on infinite objects, Handbook of theoretical computer science (van Leeuwen, Jan, editor), vol. A, Elsevier Science Publishers B.V., 1990, pp. 135–186.Google Scholar
[35] Zilber, B. I., An example of two elementarily equivalent but not isomorphic finitely generated metabelian groups, Algebra i Logika, vol. 10 (1971), pp. 309–315.Google Scholar