Hostname: page-component-76c49bb84f-ndtt8 Total loading time: 0 Render date: 2025-07-05T05:05:47.726Z Has data issue: false hasContentIssue false
  • English
  • Français

POLYCYCLIC, METABELIAN OR SOLUBLE OF TYPE (FP)∞ GROUPS WITH BOOLEAN ALGEBRA OF RATIONAL SETS AND BIAUTOMATIC SOLUBLE GROUPS ARE VIRTUALLY ABELIAN

Part of: Algebraic logic Connections with homological algebra and category theory Special aspects of infinite or finite groups

Published online by Cambridge University Press:  13 March 2017

VITALY ROMAN'KOV
VITALY ROMAN'KOV*
Affiliation:
Omsk State University n.a. Dostoevskii and Omsk State Technical University, 644077, Omsk, Russia e-mail: romankov48@mail.ru
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.

Let G be a polycyclic, metabelian or soluble of type (FP)∞ group such that the class Rat(G) of all rational subsets of G is a Boolean algebra. Then, G is virtually abelian. Every soluble biautomatic group is virtually abelian.

MSC classification

Primary: 20F16: Solvable groups, supersolvable groups
Secondary: 20J06: Cohomology of groups 03G05: Boolean algebras 20F65: Geometric group theory

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 60 , Issue 1 , January 2018 , pp. 209 - 218
Copyright
Copyright © Glasgow Mathematical Journal Trust 2017 

References

REFERENCES

1. Epstein, D. B. A., Cannon, J. W., Holt, D. F., Levy, S., Patterson, M. S. and Thurston, W., Word processing in groups (Jones and Bartlett, Boston, London, 1992).CrossRefGoogle Scholar
2. Bazhenova, G., Rational sets in finitely generated nilpotent groups, Algebra Log. 39 (4) (2000), 379–394.CrossRefGoogle Scholar
3. Gersten, S. and Short, H., Rational subgroups of biautomatic groups, Ann. Math. 134 (1991), 125–158.CrossRefGoogle Scholar
4. Gilman, R. H., Formal languages and infinite groups, in Geometric and computational perspectives on infinite groups (Minneapolis, MN and New Brunswick, NJ, 1994), DIMACS Ser. Discrete Math. Theor. Comp. Sci., vol. 25 (American Mathematical Society, Providence, RI, 1996), 27–51.Google Scholar
5. Kargapolov, M. I. and Merzlyakov, Yu. I., Foundations of the theory of groups (Springer, New York, 1979).Google Scholar
6. Kirkinskij, A. S., Intersections of finitely generated subgroups in metabelian groups, Algebra Log. 20 (1) (1981), 24–36.CrossRefGoogle Scholar
7. Kropholler, P. H., Soluble groups of type (FP)∞ have finite torsion-free rank, Bull. London Math. Soc. 25 (6) (1993), 558–566.CrossRefGoogle Scholar
8. Kropholler, P. H., Hierarchical decompositions, generalized Tate cohomology, and groups of type (FP)∞ , in Combinatorial and geometric group theory, Edinburgh, 1993 (Duncan, A.J., Gilbert, N.D. and Howie, J., Editors), London Mathematical Society Lecture Note Series, vol. 204 (Cambridge University Press, Cambridge, UK, 1995), 190–216.Google Scholar
9. Kropholler, P. H., On groups of type (FP)∞ , J. Pure Appl. Algebra 90 (1) (1993), 55–67.CrossRefGoogle Scholar
10. Lennox, J. C. and Robinson, D. J. S., The theory of infinite soluble groups (Clarendon Press, Oxford, 2004).CrossRefGoogle Scholar
11. Robinson, D. J. S., Finiteness conditions and generalized soluble groups (Springer-Verlag, Berlin, 1972).CrossRefGoogle Scholar