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.

On the Weisfeiler-Leman Dimension of Finite Groups

Given a bipartite graphG, the graphical matrix space SG consists of matrices whose non-zero entries can only be at those positions corresponding to edges in G. Tutte (J. London Math. Soc., 1947), Edmonds (J. Res. Nat. Bur. Standards Sect. B , 1967) and Lovász (FCT, 1979) observed connections between perfect matchings in G and full-rank matrices in SG. Dieudonné (Arch. Math., 1948) proved a tight upper bound on the dimensions of those matrix spaces containing only singular matrices. The starting point of this paper is a simultaneous generalization of these two classical results: we show that the largest dimension over subspaces of SG containing only singular matrices is equal to the maximum size over subgraphs of G without perfect matchings, based on Meshulam’s proof of Dieudonné’s result (Quart. J. Math., 1985). Starting from this result, we go on to establish more connections between properties of graphs and matrix spaces. For example, we establish connections between acyclicity and nilpotency, between strong connectivity and irreducibility, and between isomorphism and conjugacy/congruence. For each connection, we study three types of correspondences, namely the basic correspondence, the inherited correspondence (for subgraphs and subspaces), and the induced correspondence (for induced subgraphs and restrictions). Some correspondences lead to intriguing generalizations of classical results, such as Dieudonné’s result mentioned above, and a celebrated theorem of Gerstenhaber regarding the largest dimension of nil matrix spaces (Amer. J. Math., 1958). Finally, we show some implications of our results to quantum information and present open problems in computational complexity motivated by these results. Graduate School of Mathematics, Nagoya University, Japan (Yinan.Li@math.nagoya-u.co.jp). Research supported by MEXT Quantum Leap Flagship Program (MEXT Q-LEAP) Grant Number JPMXS0120319794. Centre for Quantum Software and Information, University of Technology Sydney, Australia (Youming.Qiao@uts.edu.au). Research supported by Australian Research Council DP200100950. School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540 (avi@ias.edu). Research supported by NSF grant CCF-1900460. Department of Mathematics, Stanford University, Stanford, CA 94305 (yuvalwig@stanford.edu). Research supported by NSF GRFP Grant DGE-1656518. Centre for Quantum Software and Information, University of Technology Sydney, Australia (Chuanqi.Zhang@student.uts.edu.au). Research supported by Australian Research Council DP200100950.

Connections between graphs and matrix spaces

The Tensor Isomorphism problem (TI) has recently emerged as having connections to multiple areas of research within complexity and beyond, but the current best upper bound is essentially the brute force algorithm. Being an algebraic problem, TI (or rather, proving that two tensors are non-isomorphic) lends itself very naturally to algebraic and semi-algebraic proof systems, such as the Polynomial Calculus (PC) and Sum of Squares (SoS). For its combinatorial cousin Graph Isomorphism, essentially optimal lower bounds are known for approaches based on PC and SoS (Berkholz&Grohe, SODA '17). Our main results are an $\Omega(n)$ lower bound on PC degree or SoS degree for Tensor Isomorphism, and a nontrivial upper bound for testing isomorphism of tensors of bounded rank. We also show that PC cannot perform basic linear algebra in sub-linear degree, such as comparing the rank of two matrices, or deriving $BA=I$ from $AB=I$. As linear algebra is a key tool for understanding tensors, we introduce a strictly stronger proof system, PC+Inv, which allows as derivation rules all substitution instances of the implication $AB=I \rightarrow BA=I$. We conjecture that even PC+Inv cannot solve TI in polynomial time either, but leave open getting lower bounds on PC+Inv for any system of equations, let alone those for TI. We also highlight many other open questions about proof complexity approaches to TI.

On the algebraic proof complexity of Tensor Isomorphism

In this paper, we explore the descriptive complexity theory of ﬁnite groups by examining the power of the second Ehrenfeucht–Fra¨ıss´e bijective pebble game in Hella’s ( Ann. Pure Appl. Log. , 1989) heirarchy. This is a Spoiler–Duplicator game in which Spoiler can place up to two pebbles each round. While it trivially solves graph isomorphism, it may be nontrivial for ﬁnite groups, and other ternary relational structures. We ﬁrst provide a novel generalization of Weisfeiler–Leman (WL) coloring, which we call 2-ary WL. We then show that the 2-ary WL is equivalent to the second Ehrenfeucht–Fra¨ıss´e bijective pebble game in Hella’s heirarchy. Our main result is that, in the pebble game characterization, only O (1) pebbles and O (1) rounds are suﬃcient to identify all groups without Abelian normal subgroups (a class of groups for which isomorphism testing is known to be in P ; Babai, Codenotti, & Qiao, ICALP 2012). In particular, we show that within the ﬁrst few rounds, Spoiler can force Duplicator to select an isomorphism between two such groups at each subsequent round. By Hella’s results ( ibid. ), this is equivalent to saying that these groups are identiﬁed by formulas in ﬁrst-order logic with generalized 2-ary quantiﬁers, using only O (1) variables and O (1) quantiﬁer depth.

On the Descriptive Complexity of Groups without Abelian Normal Subgroups

The group isomorphism problem determines whether two groups, given by their Cayley tables, are isomorphic. For groups with order $n$, an algorithm with $n^{(\log n + O(1))}$ running time, attributed to Tarjan, was proposed in the 1970s [Mil78]. Despite the extensive study over the past decades, the current best group isomorphism algorithm has an $n^{(1 / 4 + o(1))\log n}$ running time [Ros13]. The isomorphism testing for $p$-groups of (nilpotent) class 2 and exponent $p$ has been identified as a major barrier to obtaining an $n^{o(\log n)}$ time algorithm for the group isomorphism problem. Although the $p$-groups of class 2 and exponent $p$ have much simpler algebraic structures than general groups, the best-known isomorphism testing algorithm for this group class also has an $n^{O(\log n)}$ running time. In this paper, we present an isomorphism testing algorithm for $p$-groups of class 2 and exponent $p$ with running time $n^{O((\log n)^{5/6})}$ for any prime $p>2$. Our result is based on a novel reduction to the skew-symmetric matrix tuple isometry problem [IQ19]. To obtain the reduction, we develop several tools for matrix space analysis, including a matrix space individualization-refinement method and a characterization of the low rank matrix spaces.

Faster Isomorphism for p-Groups of Class 2 and Exponent p

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~$\Theta(\log n)$-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.

Abstract In recent work, Cameron, Manna and Mehatari have studied the finite groups whose power graph is a cograph, which we refer to as power-cograph groups. They classify the nilpotent groups with this property, and they establish partial results in the general setting, highlighting certain number-theoretic difficulties that arise for the simple groups of the form PSL 2 ⁡ ( q ) \operatorname{PSL}_{2}(q) or Sz ⁡ ( 2 2 ⁢ e + 1 ) \operatorname{Sz}(2^{2e+1}) . In this paper, we prove that these number-theoretic problems are in fact the only obstacles to the classification of non-solvable power-cograph groups. Specifically, for the non-solvable case, we give a classification of power-cograph groups in terms of such groups isomorphic to PSL 2 ⁡ ( q ) \operatorname{PSL}_{2}(q) or Sz ⁡ ( 2 2 ⁢ e + 1 ) \operatorname{Sz}(2^{2e+1}) . For the solvable case, we are able to precisely describe the structure of solvable power-cograph groups. We obtain a complete classification of solvable power-cograph groups whose Gruenberg–Kegel graph is connected. Moreover, we reduce the case where the Gruenberg–Kegel graph is disconnected to the classification of 𝑝-groups admitting fixed-point-free automorphisms of prime power order, which is in general an open problem.

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Classification of non-solvable groups whose power graph is a cograph

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.

A Characterization of Individualization-Refinement Trees

We prove various properties on the structure of groups whose power graph is chordal. Nilpotent groups with this property have been classiﬁed by Manna, Cameron and Mehatari [The Electronic Journal of Combinatorics, 2021]. Here we classify the ﬁnite simple groups with chordal power graph, relative to typical number theoretic oracles. We do so by devising several suﬃcient conditions for the existence and non-existence of long cycles in power graphs of ﬁnite groups. We examine other natural group classes, including special linear, symmetric, generalized dihedral and quaternion groups, and we characterize direct products with chordal power graph. The classiﬁcation problem is thereby reduced to directly indecomposable groups and we further obtain a list of possible socles. Lastly, we give a general bound on the length of an induced path in chordal power graphs, providing another potential road to advance the classiﬁcation beyond simple groups.

Jendrik Brachter

Papers

On the Weisfeiler-Leman Dimension of Finite Groups

On the Weisfeiler-Leman Dimension of Finite Groups

Classification of non-solvable groups whose power graph is a cograph

A Characterization of Individualization-Refinement Trees

On groups with chordal power graph, including a classification in the case of finite simple groups