The Weisfeiler-Leman (WL) dimension is a standard measure in descriptive complexity theory for the structural complexity of a graph. We prove that the WL-dimension of a graph on n vertices is at most 3/20The proof develops various techniques to analyze the structure of coherent configurations. This includes sufficient conditions under which a fiber can be restored up to isomorphism if it is removed, a recursive proof exploiting a degree reduction and treewidth bounds, as well as an analysis of interspaces involving small fibers.As a base case, we also analyze the dimension of coherent configurations with small fiber size and thereby graphs with small color class size.
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 derived 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.which determines how many variables are required to express a given invariant as a formula. This gives us a natural and robust framework for studying group invariants. In fact, the k-dimensional Weisfeiler-Leman algorithm is universal for all invariants of the corresponding dimension, resolving the issue of how to combine invariants. With this approach we also include an abundance of invariants that have not been considered before. However, it is a priory not clear at all that commonly used invariants can even be captured by the framework, i.e., that they even have bounded WL-dimension. ContributionThe first contribution of this paper is to show that a surprising number of isomorphism invariants and subgroups that are classic to group theory can be detected and identified by a low dimensional Weisfeiler-Leman algorithm.Specifically, we show first that for a small value of k, groups not distinguished by k-WL II have centers (k ≥ 2), inner automorphism groups (k ≥ 4), derived series (k ≥ 3), abelian radicals (k ≥ 3), solvable radicals (k ≥ 2), fitting groups (k ≥ 3) and π-radicals (k ≥ 3) that are indistinguishable by k-WL II . They also have isomorphic socles (k ≥ 5), stepwise isomorphic factors in the derived series (k ≥ 4), upper central series (k ≥ 4), and lower central series (k ≥ 4). Our techniques regarding characteristic subgroups are fairly general. We thus expect them to be applicable to a large variety of other isomorphism invariants. In particular they should facilitate the analysis of combinations of invariants one might be interested in (such as the Fitting series or the hypercenter).Beyond these characteristic subgroups, in our second contribution we show that composition factors are incorporated...
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.
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 nonisomorphism preserving transformation from graphs to groups. Using graphs of high Weisfeiler-Leman dimension, we construct highly similar but non-isomorphic groups with equal Θ(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.
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