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

This article is cited in 15 scientific papers (total in 15 papers)

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.

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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
\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

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    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  mathnet  mathscinet
    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  crossref  mathscinet  zmath  isi
    3. Blatherwick V.A., “Centraliser dimension of free partially commutative nilpotent groups of class 2”, Glasg. Math. J., 50:2 (2008), 251–269  crossref  mathscinet  zmath  isi  elib
    4. Ch. K. Gupta, E. I. Timoshenko, “Partially commutative metabelian groups: centralizers and elementary equivalence”, Algebra and Logic, 48:3 (2009), 173–192  mathnet  crossref  mathscinet  zmath  isi
    5. Kambites M., “On commuting elements and embeddings of graph groups and monoids”, Proc. Edinb. Math. Soc. (2), 52:1 (2009), 155–170  crossref  mathscinet  zmath  isi
    6. Khukhro E.I., “On solubility of groups with bounded centralizer chains”, Glasg. Math. J., 51:1 (2009), 49–54  crossref  mathscinet  zmath  isi  elib
    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  crossref  mathscinet  zmath  isi  elib
    8. Casals-Ruiz M., Kazachkov I., On systems of equations over free partially commutative groups, Mem. Amer. Math. Soc., 212, no. 999, 2011  crossref  mathscinet  isi  elib
    9. Ugurlu P., “Pseudofinite Groups as Fixed Points in Simple Groups of Finite Morley Rank”, J. Pure Appl. Algebr., 217:5 (2013), 892–900  crossref  mathscinet  zmath  isi  elib
    10. E. I. Timoshenko, “Quasivarieties generated by partially commutative groups”, Siberian Math. J., 54:4 (2013), 722–730  mathnet  crossref  mathscinet  isi
    11. E. I. Timoshenko, “Centralizer dimensions and universal theories for partially commutative metabelian groups”, Algebra and Logic, 56:2 (2017), 149–170  mathnet  crossref  crossref  isi
    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  crossref  mathscinet  isi  scopus
    13. E. I. Timoshenko, “Centralizer dimensions of partially commutative metabelian groups”, Algebra and Logic, 57:1 (2018), 69–80  mathnet  crossref  crossref  isi
    14. F. A. Dudkin, “On the centralizer dimension and lattice of generalized Baumslag–Solitar groups”, Siberian Math. J., 59:3 (2018), 403–414  mathnet  crossref  crossref  isi  elib
    15. F. A. Dudkin, “Computation of the centralizer dimension of generalized Baumslag–Solitar groups”, Sib. elektron. matem. izv., 15 (2018), 1823–1841  mathnet  crossref
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