Describing Groups

We study two complexity notions of groups - a computable Scott sentence and the index set of a group. Finding the exact complexity of one of them usually involves finding the complexity of the other, but this is not the case sometimes. J. Knight et al. determined the complexity of index sets of various structures. In this paper, we focus on finding the complexity of computable Scott sentences and index sets of various groups, generalizing methods that was previously used by J. Knight et al. We found computable Scott sentences for various different groups or class of groups, including nilpotent groups, polycyclic groups, certain solvable groups, and certain subgroups of $\mathbb{Q}$. In some of these cases, we also showed that the sentence we had are optimal. In the last section, we also give an example showing d-$\Sigma_2\subsetneq\Delta_3$ in the complexity hierarchy of pseudo-Scott sentences, contrasting the result saying d-$\Sigma_2=\Delta_3$ in the complexity hierarchy of Scott sentences, which is related to the boldface Borel hierarchy.