Book contents
- Frontmatter
- Contents
- Introduction
- Algorithms for matrix groups
- Residual properties of 1-relator groups
- Words and groups
- The modular isomorphism problem for the groups of order 512
- Recent progress in the symmetric generation of groups
- Discriminating groups: a comprehensive overview
- Extending the Kegel Wielandt theorem through π-decomposable groups
- On the prime graph of a finite group
- Applications of Lie rings with finite cyclic grading
- Pronormal subgroups and transitivity of some subgroup properties
- On Engel and positive laws
- Maximal subgroups of odd index in finite groups with simple classical socle
- Some classic and nearly classic problems on varieties of groups
- Generalizations of the Sylow theorem
- Engel groups
- Lie methods in Engel groups
- On the degree of commutativity of p-groups of maximal class
- Class preserving automorphisms of finite p-groups: a survey
- Symmetric colorings of finite groups
- References
Discriminating groups: a comprehensive overview
Published online by Cambridge University Press: 05 July 2011
Book contents
- Frontmatter
- Contents
- Introduction
- Algorithms for matrix groups
- Residual properties of 1-relator groups
- Words and groups
- The modular isomorphism problem for the groups of order 512
- Recent progress in the symmetric generation of groups
- Discriminating groups: a comprehensive overview
- Extending the Kegel Wielandt theorem through π-decomposable groups
- On the prime graph of a finite group
- Applications of Lie rings with finite cyclic grading
- Pronormal subgroups and transitivity of some subgroup properties
- On Engel and positive laws
- Maximal subgroups of odd index in finite groups with simple classical socle
- Some classic and nearly classic problems on varieties of groups
- Generalizations of the Sylow theorem
- Engel groups
- Lie methods in Engel groups
- On the degree of commutativity of p-groups of maximal class
- Class preserving automorphisms of finite p-groups: a survey
- Symmetric colorings of finite groups
- References
Summary
Abstract
Discriminating groups were introduced by Baumslag, Myasnikov and Remeslennikov as an outgrowth of their theory of algebraic geometry over groups. Algebraic geometry over groups was the main method of attack used by Kharlampovich and Myasnikov in their solution of the celebrated Tarski conjectures. The class of discriminating groups, however, has taken on a life of its own and has been an object of a considerable amount of study. In this paper we survey the large array of results concerning the class of discriminating groups that have been developed over the past decade.
Introduction
Discriminating groups were introduced by Baumslag, Myasnikov and Remeslennikov as an outgrowth of their theory of algebraic geometry over groups. Algebraic geometry over groups was the main method of attack used by O. Kharlampovich and A. Myasnikov in their solution of the celebrated Tarski conjectures. The class of discriminating groups, however, has taken on a life of its own and has been an object of a considerable amount of study. In this paper we survey the large array of results concerning the class of discriminating groups that have been developed over the past decade.
In Section 1, we define discrimination for groups and describe its ties to other areas. Also the concept of trivially discriminating (TD) groups is introduced, and the concept of squarelike groups is defined. It is also indicated how to define discrimination for arbitrary algebraic systems. It is also shown how to generalize the concept of squarelike to arbitrary algebras.
Information
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- Groups St Andrews 2009 in Bath , pp. 395 - 414Publisher: Cambridge University PressPrint publication year: 2011
References
[3] , , and , A note on nondiscrimination of nilpotent groups and Malcev completions, Contemp. Math. 421 (2006), 29–34.Google Scholar
[4] and , Reflections on some groups of B. H. Neumann, preprint.
[6] , and , Discriminating and codiscriminating groups, J. Group Theory 3 (2000), 467–479.Google Scholar
[8] , The theory of square-like abelian groups is decidable, in Algebra, logic, set theory, Stud. Log. (Lond.) 4 (Coll. Publ., London 2007), 33–46.Google Scholar
[9] and , Models and ultraproducts: an introduction (Second revised printing) (North-Holland, Amsterdam 1972).Google Scholar
[11] , private communication.
[13] , , and , Groups whose universal theory is axiomatizable by quasi-identities, J. Group Theory 5 (2002), 365–381.Google Scholar
[14] , and , Every abelian group universally equivalent to a discriminating group is elementarily equivalent to a discriminating group, Contemp. Math. 296 (2002), 129–137.Google Scholar
[15] , and , The axiomatic closure of the class of discriminating groups, Arch. Math. (Basel) 83 (2004), 106–112.Google Scholar
[16] , and , Discriminating and squarelike groups I: axiomatics, Contemp. Math 360 (2004), 35–46.Google Scholar
[17] , and , Discriminating and squarelike groups II: examples, Houston J. Math. 31 (2005), 649–673.Google Scholar
[19] , and , Discrimination properties in universal algebras, in preparation.
[20] , Subgroups of finitely presented groups, Proc. Royal Soc. London Ser. A 264 (1961), 455–475.Google Scholar
[21] , Finitely presented infinite simple groups, Notes on Pure Math. 8, (I.A.S., Austral. Nat. Univ., Canberra 1974).Google Scholar
[22] and , Groups with a quotient that contains the original group as a direct factor, Bull. Austral. Math. Soc. 45 (1992), 513–520.Google Scholar
[23] , Direct products and the Hopf property, J. Austral. Math. Soc. 17 (1974), 174–196.Google Scholar
[24] , On discriminating solvable groups, preprint.
[25] and , Irreducible affine varieties over a free group: I. Irreducibility of quadratic equations and Nullstellensatz, J. Algebra 200 (1998), 472–516.Google Scholar
[26] and , Irreducible affine varieties over a free group: II. Systems in triangular quasi-quadratic form and a description of residually free groups, J. Algebra 200 (1998), 517–569.Google Scholar
[27] and , Description of fully residually free groups and irreducible affine varieties over free groups, in Summer school in Group Theory in Banff, 1996, CRM Proc. Lecture Notes 17 (Amer. Math. Soc., Providence, RI 1999), 71–81.Google Scholar
[28] and , Hyperbolic groups and free constructions, Trans. Amer. Math. Soc. 350(2) (1998), 571–613.Google Scholar
[29] and , The elementary theory of free nonabelian groups, J. Algebra 302 (2006), 451–552.Google Scholar
[32] , Problems in combinatorial group theory, in Annals of Mathematical Studies 111 (Princeton Univ. Press 1987), 3–33.Google Scholar
[34] , On free soluble groups, Dokl. Akad. Nauk SSR 130 (1960), 495–498. English transl. Soviet Math. Dokl. (1960), 65–68.Google Scholar
[35] and , Discriminating groups and c-dimension, J. Group Theory 7(1) (2004), 135–142.Google Scholar
[38] , Diophantine geometry over groups. I. Makanin-Razborov diagrams, Publ. Math. Inst. Hautes Études Sci. 93 (2001), 31–105.Google Scholar
[39] , Diophantine geometry over groups. II. Completions, closures, and formal solutions, Israel J. Math. 104 (2003), 173–254.Google Scholar
[40] , Diophantine geometry over groups. III. Rigid and solid solutions, Israel J. Math. 147 (2005), 1–73.Google Scholar
[41] , Diophantine geometry over groups. IV. An iterative procedure for validation of a sentence, Israel J. Math. 143 (2004), 1–130.Google Scholar
[42] , Diophantine geometry over groups. V1. Quantifier elimination, Israel J. Math. 150 (2005), 1–97.Google Scholar
[43] , Diophantine geometry over groups. VI. The elementary theory of a free group, Geom. Funct. Anal. 16 (2006), 707–730.Google Scholar
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