introduce Weisfeiler-Leman-type algorithms and the notion of a WL-dimension for groups analogous to the graph case. While at first sight it seems straightforward to do so, it turns out that various concepts that coincide when applied to graphs (potentially) disagree when applied to groups. Specifically, we define three natural but different versions of a Weisfeiler-Leman dimension. One of them is based on a natural logic for groups while another is natural when taking an algorithmic viewpoint. The third version comes from natural translation of groups into graphs in an isomorphism and non-isomorphism preserving manner. Still, we give descriptions in terms of counting logics and bijective pebble games for each of the versions. A core reason why the different versions arise is that the correspondence between the various concepts arising in this context (specifically logics, algorithms and pebble games) is not as clean as for graphs. However, we argue that the definition is robust after all: we prove that the Weisfeiler-Leman dimensions of the different versions are linearly related. Overall, we obtain a family of algorithms that is similarly universal in checking combinatorial properties as in the graph case. For example, it is easy to see that the $k$-WL algorithm implicitly 