# Scott sentences for certain groups

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## Abstract

We give Scott sentences for certain computable groups, and we use index set calculations as a way of checking that our Scott sentences are as simple as possible. We consider finitely generated groups and torsion-free abelian groups of finite rank. For both kinds of groups, the computable ones all have computable \(\varSigma _3\) Scott sentences. Sometimes we can do better. In fact, the computable finitely generated groups that we have studied all have Scott sentences that are “computable *d*-\(\varSigma _2\)” (the conjunction of a computable \(\varSigma _2\) sentence and a computable \(\varPi _2\) sentence). In [9], this was shown for the finitely generated free groups. Here we show it for all finitely generated abelian groups, and for the infinite dihedral group. Among the computable torsion-free abelian groups of finite rank, we focus on those of rank 1. These are exactly the additive subgroups of \(\mathbb {Q}\). We show that for some of these groups, the computable \(\varSigma _3\) Scott sentence is best possible, while for others, there is a computable *d*-\(\varSigma _2\) Scott sentence.

## Keywords

Index sets Scott sentences Computable infinitary formulas Computable groups## Mathematics Subject Classification

03D45 03C57 20E34## Preview

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