In comparison to graphs, combinatorial methods for the isomorphism problem of finite groups are less developed than algebraic ones. To be able to investigate the descriptive complexity of finite groups and the group isomorphism problem, we define the Weisfeiler-Leman algorithm for groups. In fact we define three versions of the algorithm. In contrast to graphs, where the three analogous versions readily agree, for groups the situation is more intricate. For groups, we show that their expressive power is linearly related. We also give descriptions in terms of counting logics and bijective pebble games for each of the versions. In order to construct examples of groups, we devise an isomorphism and non-isomorphism preserving transformation from graphs to groups. Using graphs of high Weisfeiler-Leman dimension, we construct highly similar but non-isomorphic groups with equal~$\Theta(\log n)$-subgroup-profiles, which nevertheless have Weisfeiler-Leman dimension 3. These groups are nilpotent groups of class 2 and exponent~$p$, they agree in many combinatorial properties such as the combinatorics of their conjugacy classes and have highly similar commuting graphs. The results indicate that the Weisfeiler-Leman algorithm can be more effective in distinguishing groups than in distinguishing graphs based on similar combinatorial constructions.

中文翻译：

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关于有限群的 Weisfeiler-Leman 维数

与图相比，有限群同构问题的组合方法不如代数方法发达。为了能够研究有限群的描述复杂性和群同构问题，我们为群定义了 Weisfeiler-Leman 算法。事实上，我们定义了算法的三个版本。与三个类似版本很容易一致的图表相反，对于群体而言，情况更为复杂。对于群体，我们表明他们的表达能力是线性相关的。我们还对每个版本的计数逻辑和双射卵石游戏进行了描述。为了构造群的例子，我们设计了从图到群的同构和非同构保持变换。使用高 Weisfeiler-Leman 维的图，我们构造了具有相等~$\Theta(\log n)$-subgroup-profiles 的高度相似但非同构的群，但它们具有 Weisfeiler-Leman 维 3。这些群是第 2 类和指数~$p$ 的幂零群，它们在许多组合性质上是一致的，例如它们的共轭类的组合，并且具有高度相似的通勤图。结果表明，Weisfeiler-Leman 算法在区分组方面比区分基于相似组合结构的图更有效。