Universality has been an important concept in computable structure theory. A class $\mathcal{C}$ of structures is universal if, informally, for any structure of any kind there is a structure in $\mathcal{C}$ with the same computability-theoretic properties as the given structure. Many classes such as graphs, groups, and fields are known to be universal.

This paper is about the class of finitely generated groups. Because finitely generated structures are relatively simple, the class of finitely generated groups has no hope of being universal. We show that finitely generated groups are as universal as possible, given that they are finitely generated: for every finitely generated structure, there is a finitely generated group which has the same computability-theoretic properties. The same is not true for finitely generated fields. We apply the results of this investigation to quasi Scott sentences, and also answer a question of Alvir, Knight, and McCoy.

普遍性已成为可计算结构理论中的重要概念。一类$\mathcal{C}$ 如果非正式地对于任何种类的任何结构都有一个结构 $\mathcal{C}$具有与给定结构相同的可计算性-理论特性。众所周知，许多类（例如图，组和字段）是通用的。

本文是关于有限生成群的类。因为有限生成的结构相对简单，所以有限生成的组的类别没有普及的希望。我们证明了有限生成的组是尽可能通用的，因为它们是有限生成的：对于每个有限生成的结构，都有一个具有相同的可计算理论性质的有限生成的组。对于有限生成的字段，情况并非如此。我们将调查结果应用于准Scott句子，并回答了Alvir，Knight和McCoy的问题。