On the WeisfeilerLeman Dimension of Finite Groups
Abstract
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 WeisfeilerLeman 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 nonisomorphism preserving transformation from graphs to groups. Using graphs of high WeisfeilerLeman dimension, we construct highly similar but nonisomorphic groups with equal $\Theta(\sqrt{\log n})$subgroupprofiles, which nevertheless have WeisfeilerLeman 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 WeisfeilerLeman algorithm can be more effective in distinguishing groups than in distinguishing graphs based on similar combinatorial constructions.
 Publication:

arXiv eprints
 Pub Date:
 March 2020
 DOI:
 10.48550/arXiv.2003.13745
 arXiv:
 arXiv:2003.13745
 Bibcode:
 2020arXiv200313745B
 Keywords:

 Computer Science  Logic in Computer Science;
 Mathematics  Group Theory;
 68Q19;
 20D15;
 F.2.2;
 F.4.1
 EPrint:
 32 pages, 3 figures. In a previous version of the paper we calculated the orders of the groups incorrectly and therefore claimed the equality of $\Theta({\log n})$profiles rather than $\Theta(\sqrt{\log n})$profiles