Describing Groups

Two ways of describing a group are considered. 1. A group is finiteautomaton 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.

[1]  B. I. Zil'ber.  An example of two elementarily equivalent, but nonisomorphic finitely generated metabelian groups , 1971 .

[2]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[3]  A. Pillay.  Models of Peano Arithmetic , 1981 .

[4]  M. I. Kargapolov,et al.  Fundamentals of the theory of groups , 1979 .

[5]  Eitan M. Gurari,et al.  Introduction to the theory of computation , 1989 .

[6]  R. Tennant.  Algebra , 1941, Nature.

[7]  F. Oger.  Quasi-finitely axiomatizable groups and groups which are prime models , 2006 .

[8]  Anil Nerode,et al.  Automatic Presentations of Structures , 1994, LCC.

[9]  Francis Oger.  Cancellation and Elementary Equivalence of Finitely Generated Finite‐By‐Nilpotent Groups , 1991 .

[10]  Wolfgang Thomas,et al.  Automata on Infinite Objects , 1991, Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics.

[11]  Finitely Generated Groups and First‐Order Logic , 2005 .

[12]  André Nies,et al.  Finite automata presentable abelian groups , 2007, Ann. Pure Appl. Log..

[13]  Verena H. Dyson.  Review: A. I. Mal'cev, (Ob odnorn sootvetstvii mezdu kol'cami i grupparni):On a Correspondence Between Rings and Groups , 1965 .

[14]  A. Nies.  Aspects of free groups , 2003 .

[15]  G. A. Noskov.  ON THE ELEMENTARY THEORY OF A FINITELY GENERATED ALMOST SOLVABLE GROUP , 1984 .

[16]  Joaquín Pascual,et al.  Infinite Abelian Groups , 1970 .

[17]  David B. A. Epstein,et al.  Word processing in groups , 1992 .

[18]  F Oger,et al.  Quasi-finitely axiomatizable nilpotent groups , 2006 .

[19]  Anatole Khélif.  Bi-interprtabilit et structures QFA : tude de groupes rsolubles et des anneaux commutatifs , 2007 .

[20]  J. Silver,et al.  Counting the number of equivalence classes of Borel and coanalytic equivalence relations , 1980 .

[21]  Graham P. Oliver,et al.  Automatic Presentations for Finitely Generated Groups , 2005, STACS.

[22]  Achim Blumensath,et al.  Finite Presentations of Infinite Structures: Automata and Interpretations , 2004, Theory of Computing Systems.

[23]  D. Robinson.  A Course in the Theory of Groups , 1982 .

[24]  André Nies,et al.  Automatic structures: richness and limitations , 2004, Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, 2004..

[25]  M. Gromov.  Groups of polynomial growth and expanding maps , 1981 .

[26]  Chen C. Chang,et al.  Model Theory: Third Edition (Dover Books On Mathematics) By C.C. Chang;H. Jerome Keisler;Mathematics , 1966 .

[27]  Ronald Hirshon,et al.  Some cancellation theorems with applications to nilpotent groups , 1977 .