Scott sentences for certain groups
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 $\Sigma_3$ 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 $d$-$\Sigma_2$" (the conjunction of a computable $\Sigma_2$ sentence and a computable $\Pi_2$ sentence). This was already 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 $\Sigma_3$ Scott sentence is best possible, while for others, there is a computable $d$-$\Sigma_2$ Scott sentence.