RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PERSONAL OFFICE
 General information Latest issue Archive Impact factor Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Sib. Èlektron. Mat. Izv.: Year: Volume: Issue: Page: Find

 Sib. Èlektron. Mat. Izv., 2006, Volume 3, Pages 197–215 (Mi semr198)

Research papers

Centraliser dimension and universal classes of groups

Andrew J. Duncana, Ilya V. Kazatchkovb, Vladimir N. Remeslennikovc

a School of Mathematics and Statistics, University of Newcastle
b Department of Mathematics and Statistics, McGill University, Montreal, Quebec
c Omsk Branch of Mathematical Institute SB RAS

Abstract: In this paper we establish results that will be required for the study of the algebraic geometry of partially commutative groups. We define classes of groups axiomatized by sentences determined by a graph. Among the classes which arise this way we find $\mathrm{CSA}$ and $\mathrm{CT}$ groups. We study the centralisers of a group, with particular attention to the height of the lattice of centralisers, which we call the centraliser dimension of the group. The behaviour of centraliser dimension under several common group operations is described. Groups with centraliser dimension $2$ are studied in detail. It is shown that $\mathrm{CT}$-groups are precisely those with centraliser dimension $2$ and trivial centre.

Full text: PDF file (936 kB)
References: PDF file   HTML file

Bibliographic databases:

Document Type: Article
UDC: 512.54.01
MSC: 20E15, 20F10, 03B25
Received November 14, 2005, published June 7, 2006
Language: English

Citation: Andrew J. Duncan, Ilya V. Kazatchkov, Vladimir N. Remeslennikov, “Centraliser dimension and universal classes of groups”, Sib. Èlektron. Mat. Izv., 3 (2006), 197–215

Citation in format AMSBIB
\Bibitem{DunKazRem06} \by Andrew J.~Duncan, Ilya V.~Kazatchkov, Vladimir N.~Remeslennikov \paper Centraliser dimension and universal classes of groups \jour Sib. \Elektron. Mat. Izv. \yr 2006 \vol 3 \pages 197--215 \mathnet{http://mi.mathnet.ru/semr198} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2276020} \zmath{https://zbmath.org/?q=an:1118.20030} `

• http://mi.mathnet.ru/eng/semr198
• http://mi.mathnet.ru/eng/semr/v3/p197

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. J. Duncan, I. V. Kazachkov, V. N. Remeslennikov, “Orthogonal systems in finite graphs”, Sib. elektron. matem. izv., 5 (2008), 151–176
2. Vaes S., “Explicit computations of all finite index bimodules for a family of II$_1$ factors”, Ann. Sci. Éc. Norm. Supér. (4), 41:5 (2008), 743–788
3. Blatherwick V.A., “Centraliser dimension of free partially commutative nilpotent groups of class 2”, Glasg. Math. J., 50:2 (2008), 251–269
4. Ch. K. Gupta, E. I. Timoshenko, “Partially commutative metabelian groups: centralizers and elementary equivalence”, Algebra and Logic, 48:3 (2009), 173–192
5. Kambites M., “On commuting elements and embeddings of graph groups and monoids”, Proc. Edinb. Math. Soc. (2), 52:1 (2009), 155–170
6. Khukhro E.I., “On solubility of groups with bounded centralizer chains”, Glasg. Math. J., 51:1 (2009), 49–54
7. Casals-Ruiz M., Kazachkov I.V., “Elements of algebraic geometry and the positive theory of partially commutative groups”, Canad. J. Math., 62:3 (2010), 481–519
8. Casals-Ruiz M., Kazachkov I., On systems of equations over free partially commutative groups, Mem. Amer. Math. Soc., 212, no. 999, 2011
9. Ugurlu P., “Pseudofinite Groups as Fixed Points in Simple Groups of Finite Morley Rank”, J. Pure Appl. Algebr., 217:5 (2013), 892–900
10. E. I. Timoshenko, “Quasivarieties generated by partially commutative groups”, Siberian Math. J., 54:4 (2013), 722–730
11. E. I. Timoshenko, “Centralizer dimensions and universal theories for partially commutative metabelian groups”, Algebra and Logic, 56:2 (2017), 149–170
12. Ugurlu Kowalski P., “A Note on the Conjugacy Problem For Finite Sylow Subgroups of Linear Pseudofinite Groups”, Turk. J. Math., 41:6 (2017), 1458–1466
13. E. I. Timoshenko, “Centralizer dimensions of partially commutative metabelian groups”, Algebra and Logic, 57:1 (2018), 69–80
14. F. A. Dudkin, “On the centralizer dimension and lattice of generalized Baumslag–Solitar groups”, Siberian Math. J., 59:3 (2018), 403–414
15. F. A. Dudkin, “Computation of the centralizer dimension of generalized Baumslag–Solitar groups”, Sib. elektron. matem. izv., 15 (2018), 1823–1841
•  Number of views: This page: 145 Full text: 40 References: 32