Bulletin of Symbolic Logic

Describing groups

André Nies

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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.

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Bull. Symbolic Logic, Volume 13, Issue 3 (2007), 305-339.

First available in Project Euclid: 9 August 2007

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Nies, André. Describing groups. Bull. Symbolic Logic 13 (2007), no. 3, 305--339. doi:10.2178/bsl/1186666149. https://projecteuclid.org/euclid.bsl/1186666149

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