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

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.

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1 Introduction

Tasks of classifying finite groups up to isomorphism and generating particular classes of finite groups are fundamental and recurring themes in computational group theory. Yet, in particular the computational complexity of such problems remains most illusive to date.

For example, for most orders up to 20.000 the number of non-isomorphic finite groups has been computed and the groups have been exhaustively generated [13]. But there are currently 38 notoriously difficult, exceptional cases, for which this information is beyond our current means (see [13]

). The varying difficulty across different orders is in part caused by the erratic fluctuation of the number of isomorphism classes of finite groups as the order increases. This number appears to be closely linked to the multiplicities of the prime factors of the respective order, but even estimating the number of groups of a given order is non-trivial.

Generation tasks for classes of groups have a long tradition dating back to Cayley [6]. Nowadays, there is extensive work on generating particular classes of groups. For example there are practically efficient algorithms for the generation of finite nilpotent or finite solvable groups [12]. However, the algorithms come without efficient running time guarantees.

One of the difficulties for a complexity analysis stems from the group isomorphism problem. Indeed, the group isomorphism problem for finite groups stays among the few standard tasks in computational group theory with uncertain complexity. In principle, we desire algorithms with an efficient worst case running time measured in the number of generators through which the groups are given. However, we do not even have algorithms with an efficient worst case running time when measured in the order of the group. In fact the only improvement for the worst case complexity over Tarjan’s classic  algorithm are  algorithms with a small constant  depending on the model of computation (randomization, quantum computing etc.) [23, 26, 27]. There is however a nearly-linear time algorithm that solves group isomorphism for most orders [10].

A closely related problem is that of computing isomorphism invariants to distinguish groups. Efficiently computable complete invariants are sufficient for general isomorphism testing. However, we do not know efficiently computable complete invariants even for very special cases, such as nilpotent

-groups of class 2. Partial invariants only give incomplete isomorphism tests, but they still find application in generation tasks allowing for heuristic fast pruning

[13]. Given the long history of (algorithmic) group theory, there is an abundance of partial invariants.

Generally the techniques involved in generation and isomorphism computations exploit the existence of various characteristic subgroups classic to group theory. As outlined in [13], these include exploiting the Frattini subgroup  [3], the exponent--central series [25], characteristic series [28] and similar.

Overall, many of the techniques currently in use are ad-hoc, focused on practical performance, and do not lead to efficient worst case upper bounds for the complexity of the algorithmic problems. As a consequence, the general picture for finite groups is somewhat chaotic. There is often no structured way of comparing or combining invariants for group isomorphism. E.g., two given invariants may be incomparable in their distinguishing power, making it unclear which invariant to use. Also the required time to evaluate an invariant may be difficult to estimate and can depend significantly on the input group. Even when we are given a class of efficiently computable invariants, it will generally be unclear which invariants to choose or how to efficiently combine their evaluation algorithmically.

In Summary, we lack the formal means to characterize, compare, or quantify the effectiveness and complexity of invariants for group isomorphism. We therefore propose a systematic study of computationally tractable invariants for finite groups.

For inspiration on how to systematize such a study, we turn to algorithmic finite model theory and specifically descriptive complexity theory. This allows us to characterize the complexity of an invariant by considering a formula within a logic that captures the invariant. A natural choice for a logic from which to choose the formulas is the powerful fixed point logic with counting. Not only can this logic express all polynomial time computable languages on ordered structures [19, 29], but in the context of graphs it has also proven to be an effective tool in comparing invariants (see [22]). As a measure for the complexity of an invariant we can then use the number of variables required to express the invariant in fixed point logic with counting. Crucially there is a corresponding algorithm, the -dimensional Weisfeiler-Leman algorithm (WL), that (implicitly) simultaneously evaluates all invariants that are expressible by formulas requiring at most  variables in polynomial time111For groups there are actually two natural closely related versions of the logic and of the algorithm, -WL and -WL, see Section 3..

Thus, to enable a quantification and comparison of the complexity of invariants we suggest the Weisfeiler-Leman algorithm. More specifically we suggest to use the Weisfeiler-Leman dimension, 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 -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.

Contribution

The 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 , groups not distinguished by -WL have centers (), inner automorphism groups (), derived series (), abelian radicals (), solvable radicals (), fitting groups () and -radicals () that are indistinguishable by -WL. They also have isomorphic socles (), stepwise isomorphic factors in the derived series (), upper central series (), and lower central series (). 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 the invariant computed by a Weisfeiler-Leman algorithm of bounded dimension, in the following sense.

Theorem 1.1.

If and is indistinguishable from via -WL, then and have the same (isomorphism types of) composition factors (with multiplicities).

The theorem shows that the WL algorithm, which is a purely combinatorial algorithm, can compute group theoretic invariants that do not even appear as a canonical subset of the group. In particular, the composition factors cannot be localized within the group, and at first sight it might not be clear that WL grasps quotient groups.

Our third contribution, having the most technical proof and building on our other results, regards direct products of groups. Here we consider the decomposition of a group into direct factors. We show that direct products indistinguishable by -WL must arise from factors that are indistinguishable by -WL.

Theorem 1.2.

Let be a direct product and . If and  are not distinguished by -WL then there are direct factors such that and such that for all  the groups and  are not distinguished by -WL.

In other words, the Weisfeiler-Leman dimension increases by at most 1 when taking direct products. The main difficulty here is that decompositions into direct products are not unique, and thus not definable. These complications arise mainly due to central elements. However we manage to define a canonical maximal central decomposition, that is generally finer than a decomposition into direct factors. We then show that this canonical decomposition is implicitly computed by the WL algorithm.

One way of interpreting our results is that the Weisfeiler-Leman algorithm comprises a unified way of computing all the mentioned invariants and characteristics simultaneously. The dimension can therefore be used to compare the complexity of invariants.

Techniques

To show the various results on characteristic subgroups, we prove a general result on group expressions. It essentially shows that subsets that can be defined by equation systems can be detected by -WL (see Lemma 4.3).

The result on composition factors involves a technique that relates -WL distinguishability of groups to detectable normal subgroups and detectable quotients (Theorem 4.12).

To deal with direct products, we extend the technique to simultaneously relate chains of subgroups in two indistinguishable groups (Lemma 4.14). Here we exploit well-known connections of pebble games to Weisfeiler-Leman algorithms. However, the main difficulty regarding our result on decompositions into direct factors is that such decompositions are not unique. In fact in general, a group element cannot be assigned to a direct factor in a well defined sense, making it impossible for WL to detect direct factors. For this purpose we develop a new technical tool, component-wise filtrations (Definition 6.14), which compensate for the non-uniqueness to extract at least the isomorphism type of the direct factors (Lemma 6.16). We also exploit the non-commuting graph of the group and show that certain subsets, which we call non-abelian components, can be detected by -WL (Lemma 6.20). These non-abelian components lead to a WL-definable maximal central decomposition of every finite group.

Outline

Section 2 provides preliminaries. Section 3 treats WL-refinement in the context of colored groups. In Section 4, we show that invariants generated via WL-refinement fulfill group theoretic closure properties. Section 5 is an extensive collection of specific structure properties and invariants which Weisfeiler-Leman algorithms detect in finite groups. Finally, in Section 6 we investigate the ability of WL-refinement to detect direct product decompositions, building on the results of the previous sections.

Further related work

We should point out that there are various results in the literature on decomposing groups into indecomposable direct factors for various input models of groups. For example there is a polynomial time algorithm to decompose permutation groups into direct products [31]. Finally, there is a recent algorithm that finds direct product decompositions of permutation groups with factors having disjoint support [7]. There is also a polynomial time algorithm that computes direct factors efficiently for groups given by multiplication table [21]. Aspects of this algorithm are related to arguments we use for studying the behavior of WL on direct products (see the beginning of Section 6 for a discussion).

Regarding group isomorphism problems, for isomorphism of Abelian groups a linear time algorithm is known [20] and there are near linear time algorithms for some classes of non-abelian groups (e.g, [9]

). Recent directions relate group isomorphism to tensor problems

[16]. The Weisfeiler-Leman algorithm has also been incorporated as a subroutine within other sophisticated group isomorphism algorithms [5].

Regarding Weisfeiler-Leman algorithms, the literature is somewhat limited when it comes to groups [4, 5] but quite extensive when it comes to graphs. In [2], for example the authors investigate some graph invariants that are captured by the Weisfeiler-Leman algorithm. We refer to [22] for an introduction and an extensive overview over recent results for WL on graphs.

2 Preliminaries

Sets & Partitions

Maximal or minimal sets are always considered with respect to inclusion. We denote multisets as . Given disjoint sets and , their union is . An equipartition of a set is a partition such that for all . A system of representatives modulo is a subset such that for all and for all it holds that . is full if is a maximal system of representatives. We refer to the -th Cartesian power of as .

Graphs

Graphs are assumed to be undirected, simple and finite. We use and to refer to the vertices or edges of a graph . For a subset , let denote the subgraph induced by the set . A graph is called bipartite if we can write such that there are no edges in or . A matching on a graph is a collection of disjoint edges. A matching is perfect if it covers all vertices.

Groups

Groups are assumed to be finite. The symmetric group on symbols is denoted by . The order of a group element is the order of the group generated by , i.e., . Given a finite set of primes , a -group is a group whose order is only divisible by primes in . A group element is called a -element if it generates a -group. For any , let .

For a group and , we define the commutator . We abbreviate the conjugation action to . If we set and in the special case we write for the derived subgroup of .

Given -tuples of group elements , we say and have the same ordered isomorphism type if there is a group isomorphism with for all .

3 Colored Groups & Weisfeiler-Leman Algorithms

We recapitulate various notions regarding WL-algorithms on groups. For WL on graphs we refer to [22]. For uncolored groups, versions of WL were defined in [4]. For our purpose however, we need to formally generalize the concepts to the setting of colored groups. Let us point out that in the setting of colored graphs, colors can be replaced by gadget constructions to obtain uncolored graphs while maintaining the combinatorial properties of the structure. However, for groups it is unclear how to do this. Nevertheless, we will still use colors on groups to restrict the set of possible automorphisms.

3.1 Colorings on Finite Groups

Given a natural number and a finite group , a (-)coloring (over ) is just a map where denotes some finite set of colors. A -coloring partitions into color classes. We refer to -colorings as element-colorings.

The range of target colors is often omitted. Considering two natural numbers , a -coloring induces an -coloring via . To keep our notation simpler we may write again instead of and instead of we use to emphasize that the coloring is pulled back to group elements.

Definition 3.1.

A colored group is a group together with an element-coloring over . Colored groups and are isomorphic if there is a group isomorphism that respects colors, i.e., .

Given a colored group we set Aut.

Definition 3.2.

Let be a colored group. We say is -induced if it holds that , i.e., is a union of -color classes.

3.2 Weisfeiler-Leman Refinement on Colored Groups

In [4], we introduced three versions of Weisfeiler-Leman algorithms on groups. For the present work it is sufficient to consider two of these versions. The relevant definitions and results are discussed below but we refer to [4] for more details.

For we devise a Weisfeiler-Leman algorithm of dimension (-WL) that takes as input a colored group and computes an Aut-invariant coloring on . The algorithm computes an initial coloring from isomorphism invariant properties of -tuples and then iteratively refines color classes until the process stabilizes. The stable colorings arising from -WL provide (possibly incomplete) polynomial-time non-isomorphism tests.
Version I (-WL):
The initial coloring is defined via the group’s multiplication relation while also taking into account element-colors. Two tuples and obtain the same initial color if and only if for all indices and between and it holds that

1. ,

2. ,

3. .

The subsequent refinements are defined iteratively via

 χI,kγ,i+1(¯g):=(χI,kγ,i(¯g),M(¯g)).

Here, is the multiset of -tuples of colors given by

 M(¯g):={{(χI,kγ,i(¯g1←x),…,χI,kγ,i(¯gk←x))∣x∈G}},

where is obtained by replacing the -th entry of by .
Version II (-WL):
The initial coloring is defined in terms of colored, ordered isomorphism of tuples. Thus, and obtain the same initial color if and only if there exists an isomorphism of colored subgroups

 φ:⟨¯g⟩→⟨¯h⟩

such that for all . The refinement step is unchanged from Version I.

Since is finite, there is a smallest such that and induce the same color class partition on . At this point color classes become stable and we obtain the stable coloring . In the same way we define . For uncolored groups we write and , respectively.

By definition, the initial colorings are invariant under isomorphisms that respect . This property then holds for the iterated colorings as well. In particular, whenever and are isomorphic as colored groups, there is a bijection such that (and the same holds for Version II). So we obtain a non-isomorphism test by comparing stable colorings computed by -WL or -WL as follows.

Definition 3.3.

Let and be colored groups. We say is distinguished from by -WL if there is no bijection with . We say -WL identifies if it distinguishes from all other (non-isomorphic) groups. We write to indicate that and are not distinguished by -WL. Furthermore, for , tuples of group elements and are distinguished by -WL if they obtain different colors in the respective induced -colorings and . All definitions also apply to Version II in the obvious way.

The two versions of Weisfeiler-Leman refinement as introduced above are closely related and we will switch between them whenever convenient.

Lemma 3.4.

(see [4], Theorem 3.5) Let and be colored groups.

1. Consider and . If is distinguished from by -WL then is distinguished from by -WL. If is distinguished from by -WL then is distinguished from by -WL.

2. It holds

 (G,γG)≡Ik+1(H,γH)⟹(G,γG)≡IIk(H,γH)⟹(G,γG)≡Ik(H,γH).

The proof is the same as for uncolored groups, see [4] for more details. Finally, we note that in [4], we obtain a run time bound of for both versions of -WL to compute the stable coloring on . The same bound applies to colored groups. In particular, the initial coloring of -WL is efficiently computable since we only have to compute isomorphism types of -generated subgroups relative to a fixed and ordered generating set of size .

3.3 Bijective k-Pebble Games

As with graphs and uncolored groups, Weisfeiler-Leman algorithms on colored groups can be characterized via pebble games and this perspective provides useful tools for our proofs. The characterization closely follows the theory of WL-algorithms on graphs and the reader familiar with these concepts might want to skip to the next section.

For each and each version of -WL as introduced above, there is a corresponding bijective -pebble game.
Bijective -pebble game:
The -pebble game is played on a pair of colored groups of equal orders by two players called Spoiler and Duplicator. There are pairs of pebbles and pebbles from different pairs can be distinguished. A state of the game is called a configuration denoted by with and . The interpretation is that either and which means that the pebble is placed on while is placed on , or and then the -th pebble pair is currently not on the board. If we do not specify an initial configuration the game starts on the empty configuration . One round of the game consists of three steps:

1. Spoiler picks up a pebble pair .

2. Duplicator chooses a bijection .

3. Spoiler places on some and on .

In each round the winning condition is checked directly after Step . The winning condition is the only difference between the two versions of the game and it is based on the initial coloring of the corresponding version of -WL.
Version I:
The pebble pairs apart from define -tuples and over and , respectively. Spoiler wins if , where we require that there are no occurrences of in or . Otherwise the game continues.
Version II:
In this case, Spoiler wins if .

We say that Duplicator wins the game if Duplicator has a strategy to keep the game going ad infinitum.

The following correspondence between Weisfeiler-Leman refinement and pebble games is the same as in the uncolored case and can be proved in complete analogy.

Lemma 3.5.

(see [4], Theorem 3.2) Let and . Consider colored groups and with -tuples and . Then if and only if Spoiler has a winning strategy in the configuration in the -pebble game (Version ).

3.4 Induced Colorings & Refinements

Before we can start to investigate the relationship between -WL and properties of groups, we collect some useful observations on induced colorings and WL-refinement. The first lemma is well-known in the setting of graphs (or more generally for cellular algebras, see [14, Theorem 6.1]) and easily follows for groups as well.

Lemma 3.6.

Let and . Consider colored groups and with and for . Let . It holds that

1. and

2. .

Lemma 3.7.

Consider the -pebble game where for Version I and for Version II on a pair of groups . Assume pebble pairs are placed on where , and . If Duplicator chooses a bijection such that for some word (allowing inverses), then Spoiler has a winning strategy. (In the case we still require ).

Proof.

By definition of the pebble game, Duplicator chooses the bijection in Step 2 of the current round and Spoiler previously picked up a pebble pair in Step 1. Set and . In Version II, Spoiler wins by placing the pebble pair in their hands on and then picking up any pebble pair that is currently not on the board (such a pebble pair exists since ). Then the respective pebbled tuples in and have different ordered isomorphism types. So let us consider Version I.

If , then and by assumption, so Spoiler wins by pebbling and then picking up any other pebble pair. Since but , the resulting configuration is winning for Spoiler.

If , Spoiler places the pebble pair in their hands on and picks up a pebble pair that is currently not on the board. Duplicator then chooses a new bijection and without loss of generality we may assume that maps pebbled group elements accordingly (otherwise we are in the case again). Now either there is some word with and or otherwise we can write for some , such that and . In the first case, Spoiler places the pebble pair in their hands on and picks up the pebble pair on . In this case we iterate the argument. In the second case, since , up to permuting pebble pairs, Spoiler can reach the configuration

 [(gi,w′(g1,…,gn),wG,⊥,…,⊥),(hi,w′(h1,…,hn),f(wG),⊥,…,⊥)]

which fulfills the winning condition by construction of . Since the first case can only occur finitely many times, the Lemma follows. ∎

Lemma 3.8.

Let , and . Assume that there is some -tuple with -th entry such that for each with it holds that . Then -WL distinguishes from .

Proof.

We argue via Lemma 3.5, i.e., we show that Spoiler has a winning strategy in the -pebble game (Version ) with initial configuration . Duplicator’s bijection has to map to due to the initial configuration or otherwise Spoiler wins immediately. Spoiler places the -th pebble pair on . Independent of Duplicators next moves, Spoiler can subsequently pebble the entries of resulting in a configuration for some tuple with . For any such , the resulting configuration is winning for Spoiler by assumption. ∎

We say that a coloring refines a coloring , denoted , if each -color class is a union of -color classes.

Lemma 3.9.

Let be colorings on such that . Then and induce the same color classes on .

Proof.

Fix -tuples . Since holds, we also have . Assume that then by Lemma 3.6 together with the assumption . So for some we obtain and therefore . ∎

4 WL-Refinement on Quotient Groups

We are now prepared to investigate the interplay between Weisfeiler-Leman refinement and basic group structure, such as subgroups, normal closures or quotients. We introduce the notion of subset selectors to compare pairs of groups in terms of their substructures.

Definition 4.1.

A subset selector is a mapping that associates with each colored group a subset . For each version , a subset selector is called -WL-detectable, if

 χJ,kγG(S(G,γG))∩χJ,kγH(H∖S(H,γH))=∅,

for all pairs of colored groups .

When the dependency of on is clear from the context, we also say that is -WL-detectable (instead of being detectable). Examples of -WL-detectable subset selectors include the association of every group with its center () or the subset selector associating with each group the subset of elements of order .

We should remark here that in our sense detectable means that the subset of interest is a union of -color classes, but we make no statement on how to algorithmically determine which color classes form the set. In that sense it might a priory not be clear that the subset is even computable.

From the definition it follows that if is -WL-detectable then is -induced in and hence invariant under Aut. Note that defines a -WL detectable subset selector for all . Furthermore, if and are -WL-detectable then so are , and .

Definition 4.2.

A group expression of length is a sequence of subset selectors together with a set of words over variables , allowing inverses. Let be a colored group, then a -tuple is a solution to if for each it holds that and for each it holds that . Let denote the set of all solutions to over .

Lemma 4.3.

Consider a group expression . Let and assume that each is -WL-detectable.

1. Let and be colored groups. Then all -tuples in can be distinguished from all -tuples in via -WL.

2. For and colored groups define

 Sol∃j(G,γ) :={x∈G∣∃(x1,…,xt)∈SolE(G,γ):xj=x} Sol∀j(G,γ) :={x∈G∣(∀xi∈Si(G,γ))1≤i≤t:(x1,…,xj−1,x,xj+1,…,xt)∈SolE(G,γ)}.

Then and are -WL-detectable subset selectors for all .

The same holds for -WL, provided .

Proof.
1. Let and . First consider the case that there is some word such that . Then there does not exist an isomorphism between and that maps to for all . Thus, by definition, the -tuples and obtain different initial colors in -WL. In the other case, since , there must be some index such that . By assumption is detectable by -WL and , so by definition . In particular, and can be distinguished via Lemma 3.6.

The proof for -WL is almost identical. Note that for Version I we assume that . Then, in the first case where the tuples fulfill different relations we use Lemma 3.7 to obtain the result for -WL. The second case can be treated identically for both Version I and Version II.

2. Using the statement from Part 1, if we consider and we are exactly in the situation of Lemma 3.8 and so and can be distinguished via -WL, i.e., is detectable. The same argument works for and . The latter is equivalent to the existence of such that , so in this situation we use Lemma 3.8 for . ∎

Lemma 4.4.

Consider -WL-detectable subset selectors . Then the following subset selectors are -WL-detectable:

1. for each , where ,

2. , where .

Provided is at least , -WL further detects the following subset selectors:

1. for each , in particular also ,

2. , in particular also ,

3. , where ,

4. , where .

All statements remain true if we replace Version II by Version I everywhere (including the assumptions), provided in Parts 1 and 2 and in Parts 3–6.

Proof.

We make repeated use of Lemma 4.3. Given a group expression , define and ( as in Lemma 4.3.

1. Set . Then .

2. Set . Then .

3. We argue by induction over . Let us write . Assume that can be detected for and consider . Then is exactly and since and are both detectable, so is . In particular, is detectable by -WL as a union of detectable subset selectors.

4. Set . The -conjugates of elements in are precisely . Together with Part 3, this shows that the normal closure of is detectable by -WL for .

5. Set . If is detectable then so is which implies that is detectable. Finally note that elements of do not normalize if and only if they belong to .

6. Set . Then is the set of all --commutators and using Part 3, we obtain detectability of the group they generate, namely .

The analogue statements for -WL follow from Lemma 4.3 as well, provided in each case. ∎

We highlight two direct implications of the previous lemma.

Corollary 4.5.

Let , and . If then

 (⟨xG⟩,γG|⟨xG⟩)≡IIk−1(⟨yH⟩,γH|⟨yH⟩).

The same holds for Version I with .

Corollary 4.6.

Let , and for . If then is normal in if and only if