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Describing Groups

Published online by Cambridge University Press:  15 January 2014

André Nies
André Nies*
Affiliation:
Department of Computer Science, Auckland University, Auckland, New Zealand, E-mail: [email protected]
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Abstract

Two ways of describing a group are considered. 1. A group is finite-automaton 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.

Type
Articles
Information
Bulletin of Symbolic Logic , Volume 13 , Issue 3 , September 2007 , pp. 305 - 339
Copyright
Copyright © Association for Symbolic Logic 2007

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References

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