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DOI: 10.4230/LIPIcs.ESA.2022.27

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

Authors: Jendrik Brachter and Pascal Schweitzer

Published in: LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

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.

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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)

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@InProceedings{brachter_et_al:LIPIcs.ESA.2022.27,
  author =	{Brachter, Jendrik and Schweitzer, Pascal},
  title =	{{A Systematic Study of Isomorphism Invariants of Finite Groups via the Weisfeiler-Leman Dimension}},
  booktitle =	{30th Annual European Symposium on Algorithms (ESA 2022)},
  pages =	{27:1--27:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-247-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{244},
  editor =	{Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.27},
  URN =		{urn:nbn:de:0030-drops-169653},
  doi =		{10.4230/LIPIcs.ESA.2022.27},
  annote =	{Keywords: group isomorphism problem, Weisfeiler-Leman algorithms, group invariants, direct product decompositions}
}

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DOI: 10.4230/LIPIcs.ISAAC.2021.24

A Characterization of Individualization-Refinement Trees

Authors: Markus Anders, Jendrik Brachter, and Pascal Schweitzer

Published in: LIPIcs, Volume 212, 32nd International Symposium on Algorithms and Computation (ISAAC 2021)

Abstract

Individualization-Refinement (IR) algorithms form the standard method and currently the only practical method for symmetry computations of graphs and combinatorial objects in general. Through backtracking, on each graph an IR-algorithm implicitly creates an IR-tree whose order is the determining factor of the running time of the algorithm. We give a precise and constructive characterization which trees are IR-trees. This characterization is applicable both when the tree is regarded as an uncolored object but also when regarded as a colored object where vertex colors stem from a node invariant. We also provide a construction that given a tree produces a corresponding graph whenever possible. This provides a constructive proof that our necessary conditions are also sufficient for the characterization.

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Markus Anders, Jendrik Brachter, and Pascal Schweitzer. A Characterization of Individualization-Refinement Trees. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 24:1-24:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{anders_et_al:LIPIcs.ISAAC.2021.24,
  author =	{Anders, Markus and Brachter, Jendrik and Schweitzer, Pascal},
  title =	{{A Characterization of Individualization-Refinement Trees}},
  booktitle =	{32nd International Symposium on Algorithms and Computation (ISAAC 2021)},
  pages =	{24:1--24:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-214-3},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{212},
  editor =	{Ahn, Hee-Kap and Sadakane, Kunihiko},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2021.24},
  URN =		{urn:nbn:de:0030-drops-154578},
  doi =		{10.4230/LIPIcs.ISAAC.2021.24},
  annote =	{Keywords: individualization refinement algorithms, backtracking trees, graph isomorphism}
}

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Documents authored by Brachter, Jendrik

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

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A Characterization of Individualization-Refinement Trees

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