groups. However, depending on the interpretation of these methods, we will obtain several different choices for initial colorings and refinement strategies for finite groups. We will argue that different methods are all in some sense natural and interesting in their own right. However, the different concepts (possibly) lead to different notions of Weisfeiler-Leman dimension for groups. In contrast to this, for graphs, all notions are equivalent. While we are not able to precisely determine whether for groups the different methods are equally powerful at this point, we do however show that exchanging one method for another changes the dimension by at most a constant factor.

Weisfeiler-Leman algorithms for groups
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The following algorithms define color-refinement procedures on $k$-tuples of group elements. Since groups are (essentially) ternary relational structures, we will usually require for the dimension that $k\geq 2$. In the following let $G$ be a group. We will define three versions of the WL-algorithm for groups.

#### Version I:

Define an initial coloring $\chi_0\colon G^k \rightarrow C$ on $k$-tuples of group elements so that $(g_1,\dots,g_k)$ and $(h_1,\dots,h_k)$ obtain the same color if and only if for all indices $i,j\in \{1,\ldots,k\}$ we have $g_i=g_j$ exactly if $h_i=h_j$ and for all indices 