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A Systematic Study of Isomorphism Invariants of Finite Groups via the Weisfeiler-Leman Dimension

Authors Jendrik Brachter, Pascal Schweitzer

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Author Details

Jendrik Brachter
  • TU Darmstadt, Germany
Pascal Schweitzer
  • TU Darmstadt, Germany

Funding

The research leading to these results has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (EngageS: grant agreement No. 820148) and from the German Research Foundation DFG (SFB-TRR 195 "Symbolic Tools in Mathematics and their Application").

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Jendrik Brachter and Pascal Schweitzer. A Systematic Study of Isomorphism Invariants of Finite Groups via the Weisfeiler-Leman Dimension. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 27:1-27:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.ESA.2022.27

Abstract

We investigate the relationship between various isomorphism invariants for finite groups. Specifically, we use the Weisfeiler-Leman dimension (WL) to characterize, compare and quantify the effectiveness and complexity of invariants for group isomorphism.
It turns out that a surprising number of invariants and characteristic subgroups that are classic to group theory can be detected and identified by a low dimensional Weisfeiler-Leman algorithm. These include the center, the inner automorphism group, the commutator subgroup and the derived series, the abelian radical, the solvable radical, the Fitting group and π-radicals. A low dimensional WL-algorithm additionally determines the isomorphism type of the socle as well as the factors in the derives series and the upper and lower central series.
We also analyze the behavior of the WL-algorithm for group extensions and prove that a low dimensional WL-algorithm determines the isomorphism types of the composition factors of a group.
Finally we develop a new tool to define a canonical maximal central decomposition for groups. This allows us to show that the Weisfeiler-Leman dimension of a group is at most one larger than the dimensions of its direct indecomposable factors. In other words the Weisfeiler-Leman dimension increases by at most 1 when taking direct products.

Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity theory and logic
  • Mathematics of computing → Combinatorial algorithms
Keywords
  • group isomorphism problem
  • Weisfeiler-Leman algorithms
  • group invariants
  • direct product decompositions

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