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 $\Theta(\sqrt{\log n})$-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.
翻译:与图表相比, 定数组的异形问题分类方法比代数问题更不完善。 为了能够调查定数组的描述复杂性和集团的异形问题, 我们定义了集团的 Weisfeiler- Leman 算法。 事实上, 我们定义了算法的三个版本。 与图表相比, 三个相似版本很容易达成一致, 情况对于集团来说非常相似, 我们显示它们的表达力是线性关联的。 我们还给出了每个版本的逻辑和双向方方形游戏的描述。 为了构建群的示例, 我们设计了一种异形和非异形法来保护从图表到群的转变。 使用高 Weisfeiler- Leman 维度的图表图, 我们构建了非常相似但非异形的组, 与等值的 $( sqirtrt) 相近。 对于集团, 我们显示的是 Weisfeiler- Leman 的表达力和双向方形方形方形游戏。 这些组是第2类的零级组, 和直径方形的直径方形图类的组, 它们在高方形图类的类中比较相似的类的直径直径直径直径直方形的类中, 。 它们在高直形的类的类的类的类的直形图级的类中, 它们在以直径直径等的直径直径直系的直径可见的直径等的直系的直形结构的直图显示的直径, 。