Paper Info
Title
On the Weisfeiler-Leman 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 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 similar but non-isomorphic groups with equal ™(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.
Year
DOI
Venue
2020
10.1145/3373718.3394786
LICS '20: 35th Annual ACM/IEEE Symposium on Logic in Computer Science Saarbrücken Germany July, 2020
Keywords
DocType
ISSN
group isomorphism, Weisfeiler-Leman, logics with counting, Cai-Furer-Immermann graphs
Conference
1043-6871
ISBN
Citations 
PageRank 
978-1-4503-7104-9
0
0.34
References 
Authors
0
2
Name
Order
Citations
PageRank
Brachter Jendrik100.34
Pascal Schweitzer221416.94