Describing groups

We study two complexity notions of groups - the syntactic complexity of a computable Scott sentence and the $m$-degree of 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 always the case. 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. We give computable Scott sentences for various different groups, including nilpotent groups, polycyclic groups, certain solvable groups, and certain subgroups of $\mathbb{Q}$. In some of these cases, we also show that the sentences we give are optimal. In the last section, we also show that d- $\Sigma _2\subsetneq \Delta _3$ in the complexity hierarchy of pseudo-Scott sentences, contrasting the result saying d- $\boldsymbol {\Sigma }_2=\boldsymbol {\Delta }_3$ in the complexity hierarchy of Scott sentences, which is related to the boldface Borel hierarchy.