1 Introduction
The notion of isomorphisms between finite groups remains one of the most basic concepts of group theory for which we do not have efficient algorithmic tools. The algorithmic Group Isomorphism Problem formalizes the task of deciding whether two given (finite) groups are isomorphic, but in fact, we do not understand its complexity. We have neither a polynomial time algorithm for testing isomorphism, nor complexity theoretic evidence indicating to us that the problem is not polynomial time solvable. Considering groups of order , a simple approach, attributed to Tarjan in [28], is to pick a small generating set in one of the groups and to check for all possible images of the generators in the other group, whether the partial map extends to an isomorphism. This approach gives us a worstcase runtime of where is the size of the generating set. Since every group of order has a generating set of size at most , this yields in the worst case. Despite decades of active research this bound has seen only slight improvements for the general case. In fact, Rosenbaum [33] was able to improve it to . (See [26] and [34] for related discussions on isomorphism of groups and solvable groups). For various classes of groups, better bounds are known (see further related work). However, even very limited classes of groups provide hard cases for isomorphism testing. One of the most prominent classes in this context is formed by the groups of prime exponent and nilpotency class . Such groups possess a lot of extra structure, but despite this and despite a large body of research into this structure, even for this limited class, no better general bound has been proven. In fact, this class seems to be at the core of the problem. However, a formal reduction to this or a similar class is not known.
While there exists a vast collection of algebraic methods and heuristics for tackling the group isomorphism problem (see further related work), complexity theoretic and combinatorial aspects seem to be less developed. For example, in 2011, Timothy Gowers asked on Lipton’s blog
[16] whether there is an integer such that the isomorphism class of each finite group is determined by their subgroupprofile. Here the subgroupprofile (or profile) is the multiset of (isomorphism types of) generated subgroups. Glauberman and Grabowski gave a negative answer by constructing pairs of nonisomorphic groups with the same profiles [14]. Subsequently, Wilson constructed many examples of exponent and nilpotency class groups which agree in various invariants. In particular they have the same profiles [39], which is best possible.The observation that combinatorial aspects of the group isomorphism problem are less developed is surprising since, for the related graph isomorphism problem, historically, it has been the other way around. Indeed, for graph isomorphism testing, combinatorial approaches are welldeveloped and often successful, yet their limits have been firmly established. One of the most important tools in this scope is the WeisfeilerLeman algorithm. The dimensional WeisfeilerLeman algorithm (
WL) iteratively classifies
tuples of vertices of a graph in terms of how they are related to other vertices in the graph. It provides an effective invariant for graphnonisomorphism (see e.g. [36, 7]). Moreover, WL can be implemented to run in time where is the number of vertices (see [21, 22]). For fixed , the algorithm is only a partial isomorphism test, in that it can distinguish certain pairs of nonisomorphic graphs, but not all of them. A graph is said to have WLdimension at most , if WL distinguishes the graph from every nonisomorphic graph. For many important classes of graphs the WLdimension has been shown to be bounded; examples include planar graphs [18, 24] for which even suffices and more generally classes defined by forbidden minors [19]. On the other hand, Cai, Fürer and Immerman constructed an infinite family of graphs, for which the WLdimension is linear in the number of vertices, and thus unbounded [7]. Higher dimensional versions of WL also appear in Babai’s breakthrough result putting graphisomorphism in quasipolynomial time [1].There is a deep and wellunderstood connection between WL and the expressiveness in the logic , the extension of the variable fragment of first order logic on graphs with counting quantifiers [7]. For example, two graphs can be distinguished by WL exactly if there is a formula in that distinguishes the graphs. Therefore, in some welldefined sense, the WL algorithm is universal in that it simultaneously checks all combinatorial properties in an input graph expressible in the aforementioned logic.
Contribution. The first aim of this paper is to introduce WeisfeilerLemantype algorithms and the notion of a WLdimension for groups analogous to the graph case. While at first sight it seems straightforward to do so, it turns out that various concepts that coincide when applied to graphs (potentially) disagree when applied to groups. Specifically, we define three natural but different versions of a WeisfeilerLeman dimension. One of them is based on a natural logic for groups while another is natural when taking an algorithmic viewpoint. The third version comes from natural translation of groups into graphs in an isomorphism and nonisomorphism preserving manner. Still, we give descriptions in terms of counting logics and bijective pebble games for each of the versions. A core reason why the different versions arise is that the correspondence between the various concepts arising in this context (specifically logics, algorithms and pebble games) is not as clean as for graphs. However, we argue that the definition is robust after all: we prove that the WeisfeilerLeman dimensions of the different versions are linearly related. Overall, we obtain a family of algorithms that is similarly universal in checking combinatorial properties as in the graph case. For example, it is easy to see that the WL algorithm implicitly computes the profile of groups. In particular, abelian groups are completely identified already by the least powerful of the algorithms.
The second aim of this paper is to understand when and how groups are characterized by their combinatorial properties. On the one hand this addresses the question whether combinatorial methods can solve the Group Isomorphism Problem. On the other hand it provides a way of quantifying similarity of nonisomorphic groups. Specifically, we construct pairs of arbitrarily large nonisomorphic groups that agree with respect to many isomorphism invariants but can still be distinguished with the dimensional WLalgorithm. More precisely these groups are of nilpotency class and prime exponent . They are nonisomorphic but have the same profile. They also have highly similar commuting graphs.
Theorem 1.1.
For infinitely many there exist pairs of nonisomorphic groups order with bounded WeisfeilerLeman dimension which

have equal profiles,

have commuting graphs that are indistinguishable for the dimensional WeisfeilerLeman algorithm (for graphs),

are of exponent and nilpotency class , and

have equal sizes of conjugacy classes.
The theorem shows that the WeisfeilerLeman algorithm can be more effective in distinguishing groups than in distinguishing graphs even when they are based on similar combinatorial constructions. The proof that the WLdimension is low intuitively indicates that the ability to fix products of elements appears to be related to the ability to fix sets of elements and how to exploit this.
In comparison to the previous constructions mentioned above, our construction has the advantage that it is of a purely combinatorial nature. It is therefore easy to analyze the groups, and many combinatorial properties of the resulting groups can be tuned. In fact, we can start with an arbitrary graph and encode it into a group while preserving isomorphisms. We should stress that even though we start with graphs of unbounded WeisfeilerLeman dimension, the resulting groups have only dimension 3. This highlights the power of WeisfeilerLemantype algorithms to distinguish groups beyond the scope of traditional invariants.
1.1 Further related work
Our work can be understood as studying the descriptive complexity of finite groups. We refer to Grohe’s monograph [19] for extensive information on the descriptive complexity of graphs (rather than groups). A central result in [19] shows graph classes with a forbidden minor have bounded WLdimension. A recent paper relating first order logics and groups is [29]. The descriptive complexity of finite abelian groups has been studied in [15]. However, descriptive complexity of groups has been investigated considerably less than that of graphs. In contrast to this, the research body on the algorithmic Group Isomorphism Problem is extensive. The results can generally be divided into research with a more practical and research with a more theoretical focus.
On the practical side the best algorithms for isomorphism testing are typically implemented in computer algebra systems such as SAGE, MAGMA, GAP. Classical algorithms include the one by Smith [35] (for solvable groups), the one by Eick, LeedhamGreen and O’Brien (for groups) [12, 30], as well as a general algorithm by Cannon and Holt [9]. Newer algorithms have been developed by Wilson [38] with numerous improvements over time together with Brooksbank and Maglione [6]. More recent work introduces ever stronger invariants to distinguish groups quickly. We refer to [5] for an overview and the most recent techniques and an algorithm incorporating many of them. Dietrich and Wilson report that current isomorphism tests are already infeasible in practice on some groups with orders in the thousands [11].
In any case, in our work we focus on the theoretical side. As mentioned before, the best bound for the general problem is by Rosenbaum [33]. Polynomial time algorithms have been developed for various classes of groups [2, 3, 4, 10, 13, 17, 23, 32]. There is an algorithm running in polynomial time for most orders [11]. For the currently fastest isomorphism algorithm for permutation groups see [37].
Recent efforts incorporate the WeisfeilerLeman algorithm into the group isomorphism context [5, 25]
. However, there is a crucial difference to our work. Indeed, in these papers the authors use a combinatorial construction within the groups acting on vector spaces on which the (graph) WLalgorithm is executed. This is different to the general algorithm for all groups defined here. Thus, a priori the two algorithmic approaches are unrelated, warranting further study.
2 Preliminaries and notation
Groups.
Groups will be denoted by capital Latin characters. For a group and elements we write their commutator as and we use to refer to the subgroup of generated by all commutators. Then is the unique minimal normal subgroup of with abelian quotient. The centralizer of is and then is the center of . For a prime , a group is called a group if is a power of (in particular, we assume to be finite here). The exponent of a group is the least common multiple of the orders of its elements. A group is elementary abelian if it is abelian and of prime exponent (i.e., for some ). The Frattinisubgroup of a group is the intersection of all maximal subgroups. If is a group then is the unique minimal normal subgroup of with elementary abelian quotient. The elements of are nongenerators in , that is, if generates then so does .
We define a fold commutator to be just a regular commutator and then a fold commutator is an element of the form with and a fold commutator in . A group is said to be nilpotent if there is some integer such that fold commutators are always trivial in and if this is the case then the nilpotency class of is the smallest such . For example abelian groups are exactly the groups of nilpotency class and a group has nilpotency class if and only if it is nonabelian and every commutator is central.
A group isomorphism is a bijective map that preserves group multiplication. We collect all isomorphisms between and in a (possibly empty) set and set . We write for the set of all subgroups of .
We assume the term ’group’ to mean ’finite group’ and whenever we include infinite groups we do so explicitly.
Graphs.
All graphs will be finite simple undirected graphs and referred to with greek characters, primarily , subject to suitable subscripts. That is, a graph consists of a finite set of vertices and a set of edges . The complement of will always be the simple complement graph, namely . The set of neighbors of is and is the closed neighborhood of . The degree of is . A graph is regular if every vertex has degree . For a set of vertices , the induced subgraph is .
An isomorphism of graphs is a bijective map that simultaneously preserves edges and nonedges. The set of isomorphisms between and is and define .
The commuting graph of a group is the graph whose vertices are the group elements and two distinct elements are adjacent if .
2.1 The WLalgorithm for graphs
Before we explore how the WLalgorithm can be applied to groups, we briefly recapitulate its classic definition for graphs. Given a graph , the dimensional version of the algorithm for positive repeatedly colors the tuples of vertices with abstract colors that encode how each tuple is situated within the graph. The initial coloring of each tuple encodes the isomorphism type of the graph induced by , taking into account where the vertices occur in the tuple. Specifically, a coloring into some set of colors is defined so that holds exactly if there is an isomorphism from to which sends to for all .
The coloring is now iteratively refined as follows. For a tuple and , define to be the tuple obtained by replacing the th entry with . Then we define for the coloring
Here denotes multisets. Thus the color of the next iteration consists of the color of the previous iteration and the multiset of colors obtained by replacing each entry in the tuple with another vertex from the graph. For the definition is slightly different, namely that for the multiset is only taken over vertices in the neighborhood .
Adding the color of the previous iteration as first entry ensures that the partition induced on by is finer than (or as fine as) the partition induced by . Let be the least positive integer for which the partition induced by agrees with the partition induced by , then we define the final coloring to be . Since the domain of the has size , we know that . For fixed , it is possible to compute the partition of in time [21].
To distinguish two nonisomorphic graphs the algorithm is applied on the disjoint union. If in the final coloring the multiset of colors appearing in one graph is different than those appearing in the other graph, then the graphs are not isomorphic. The converse does not necessarily hold, as we explain next.
2.2 The CFIgraphs
As mentioned previously, for each there is a pair of nonisomorphic graphs not distinguished by WL.
Theorem 2.1 (Cai, Fürer, Immerman [7]).
There is an infinite family of pairs of nonisomorphic 3regular graphs on vertices not distinguished by the dimensional WeisfeilerLeman algorithm.
Since we intend to exploit the construction by transferring it to groups, we describe it next. We start with a connected base graph . In this graph every vertex is replaced by a particular gadget and the gadgets are interconnected according to the edges of as follows. For a vertex of degree we use the gadget , which is a graph whose vertex set consists of external vertices and internal vertices . The internal vertices form a copy of the set of those 01strings of length that have an even number of entries equal to 1. For each , each internal vertex is adjacent to exactly one vertex of , namely it is adjacent to if the th bit of the string is 0 and to otherwise. An example of is depicted in Figure 1. It remains to explain how the different gadgets are interconnected. For this, for a vertex of degree each edge is associated with one of the pairs . For an edge assume is associated with the pair in the gadget corresponding to and is associated with the pair in the gadget corresponding to . Then we insert (parallel) edges and . Adding such parallel edges for each edge of the base graph we obtain the graph . The twisted CFIgraph is obtained by replacing one pair of (parallel) edges and with the (twisted) edges and . It can be shown that for connected base graphs (up to isomorphism) it is irrelevant which edge is twisted [7]. For a subset of the edges of the base graph , we can define the graph obtained by twisting exactly the edges in . The resulting graph is isomorphic to if is even and isomorphic to otherwise.
In the original construction the base graph is usually thought of as vertex colored with all vertices obtaining a different color. This makes all gadgets distinguishable. The colors can be removed by attaching gadgets retaining the property that the base graph is identified by 2dimensional WeisfeilerLeman. We want to record here the observation that it is possible to choose the base graph of WLdimension 2 while maintaining the property that it is 3regular.
Observation 2.2.
The 3regular base graph can be chosen to have WeisfeilerLeman dimension at most 2.
This can be seen in two ways, by adding gadgets on edges or by observing that random expanders, usually used in the construction, have this property.
2.3 First order logic with counting
There is a close connection between the WeisfeilerLeman algorithm of dimension and the variable fragment of first order logic on graphs with counting quantifiers [7]. To obtain this logic we endow first order logic with counting quantifiers. The formula expresses then the fact that there are at least distinct elements that satisfy the formula . For example the formula would express that the graph contains at least 3 vertices of degree at least 4. The logic is the fragment of said logic which allows formulas to only use distinct variables (that can however be reused an arbitrary number of times). We refer to [21] for a more thorough introduction to these logics and a proof that two graphs can be distinguished by dimensional WL exactly if there is a formula in that holds on the one graph but not on the other. Often such logics are endowed with a fixedpoint operator, but since we will only apply the formulas to structures of fixed size, this will not be necessary for us (see [31] for more information).
2.4 The pebble game
There is a third concept, the bijective pebble game [20], that has a deeper connection to the logic and the WL. This game is often used to show that graphs cannot be distinguished by WL. The game is an EhrenfeuchtFraïssétype game with two players Duplicator and Spoiler. Initially pairs of pebbles, each pair uniquely colored, are placed next to two given input graphs . Each round proceeds as follows: Spoiler picks up a pebble pair of pebbles of the same color. Then Duplicator chooses a bijection from to . Then Spoiler places pebble on a vertex and places on . Spoiler wins if at any point in time the graph induced by the vertices occupied by pebbles in is not isomorphic to the graph induced by the vertices occupied by pebbles in via a map that sends a pebble to its corresponding pebble of the same color in the other graph. Spoiler also wins (in round 0) if .
When using pebbles on two graphs, the game can be won by Spoiler exactly if WL distinguishes the graphs [20].
3 WLtype algorithms on groups
As with graphs we would like to be able to study combinatorial properties of finite groups using WLtype algorithms. The natural approach is to adapt the methods from the last section to suit (finite) groups. However, depending on the interpretation of these methods, we will obtain several different choices for initial colorings and refinement strategies for finite groups. We will argue that different methods are all in some sense natural and interesting in their own right. However, the different concepts (possibly) lead to different notions of WeisfeilerLeman dimension for groups. In contrast to this, for graphs, all notions are equivalent. While we are not able to precisely determine whether for groups the different methods are equally powerful at this point, we do however show that exchanging one method for another changes the dimension by at most a constant factor.
3.1 WeisfeilerLeman algorithms for groups
The following algorithms define colorrefinement procedures on tuples of group elements. Since groups are (essentially) ternary relational structures, we will usually require for the dimension that . In the following let be a group. We will define three versions of the WLalgorithm for groups.
Version I:
Define an initial coloring on tuples of group elements so that and obtain the same color if and only if for all indices we have exactly if and for all indices we have exactly if . We iteratively define the refinement in the classical way just like it is defined for graphs, that is for we have
Version II:
In the definition for graphs, the initial coloring of a tuple takes into account the subgraph induced by the tuple. In analogy to this, one might argue that for groups the initial coloring needs to take into account the subgroup generated by the tuple. Thus, in Version II, we define an initial coloring on tuples of group elements such that and obtain the same color if and only if there is a map with which extends to an isomorphism from to . In this case we will also say the tuples agree in their marked isomorphism type. The iterative refinement is again performed in the classical way.
Version III:
For Version III, we encode groups as graphs, execute the WLalgorithm for graphs and pull back the coloring. For this choose an isomorphismpreserving, invertible functor that maps finite groups to finite (simple) graphs . Our working example will be the following construction but other choices are certainly possible and many choices will lead to equivalent or at least related algorithms.
To obtain the graph (see Figure 2), start with a set of isolated nodes corresponding to elements of the group . For each pair of group elements add a multiplication gadget by adding 4 nodes and add the edges
We then use the classical dimensional WLalgorithm on the graph and pull back the colorings of tuples by simply restricting it to .
By construction we have and due to vertexdegrees is a canonical subset of the vertices of . Thus for two groups we have if and only if .
Many other reductions of this form transforming groups to graphs are possible. However, some of them are artificial. For example, one could artificially ensure that the resulting graph has low WLdimension by precomputing certain isomorphisminvariants not captured by the WLalgorithm. Thus, a unified treatment of all isomorphismpreserving constructions seems infeasible. However, it seems that for many ’wellbehaved’ functors the WLdimension of the constructed graphs differ by a constant factor only. On another note, it would be interesting to obtain efficiently computable subquadratic reductions from groups to graphs, but we are not aware of such a construction.
For fixed , each version of the WLalgorithm gives rise to a polynomialtime (possibly) partial isomorphism test on pairs of finite groups. Indeed, marked isomorphism of tuples in a group can be checked in time so we can compute initial colors in time for Versions I and II. The refinement steps are the same as for graphs and thus we obtain the same bound for the rest of the computation. For Version III we have a quadratic blowup yielding time .
Definition 3.1.
Groups and are equivalent with respect to WL in Version , in symbols , if there is a bijection preserving final colors of the respective colorrefinement procedure. Furthermore we write if it holds that , i.e., the distinguishing power of is weaker than or equal to the distinguishing power of .
The main result of this section is that we can exchange one version for another when we multiply the dimension with a constant factor. When studying WLtype algorithms it is often useful to have equivalent pebble games at hand, so we first associate a pebble game to each of the variants above.
3.2 Bijective pebble games
We now define suitable pebble games for the different versions. Each of these games is played by two players Spoiler and Duplicator and in each case we will say that Duplicator wins the game if and only if there is a strategy for Duplicator to keep the game going on forever. The board consists of a pair of finite groups of equal order (or rather their elements) or a pair of corresponding graphs for Version III, respectively. There are pairs of pebbles . We think of pebbles in the same pair as having the same color, and pebbles from different pairs as having distinct colors. The pebbles can be placed beside the board or on the group elements (graph vertices in Version III), in which case we say a group element is pebbled. Pebbles are placed on elements of (vertices of ) and pebbles on (). At any point in time the pebbles give us a pebbled tuple in (or ), where indicates that the pebble is placed besides the board.
Version I:
All pairs of pebbles are initially placed beside the board. A round of the game consists of these steps:

Spoiler picks up a pair of pebbles .

Duplicator chooses a bijection .

Spoiler pebbles some element with , the corresponding pebble is placed on .
The winning condition is always checked right after Step 1. At that moment, the pebbles not in Spoiler’s hand then pebble a
tuple over and a corresponding tuple over . Spoiler wins if the pebbled tuples differ with respect to the initial coloring of Version I. (This implies that no more pebbles are placed beside the graph.)Version II:
Version II differs from Version I only in that the winning condition uses the initial coloring of Version II rather than Version I. That is, Spoiler wins if the map induced by pairs of pebbles does not extend to an isomorphism between the subgroups generated by the pebbled group elements, and the game continues otherwise.
Version III:
Version III is the (classical) bijective pebble game for graphs played on and (see Subsection 2.4).
In the pebble games, when we say that “Duplicator has to do something”, we mean that otherwise Spoiler wins the game. We say that Duplicator respects a certain property of group elements if Duplicator always has to pebble pairs of groups elements which agree in whether they have the property. One can show that Duplicator must respect the partial mapping given by the pairs of pebbles that are currently on the board. Indeed, otherwise Spoiler can win in the next round by pebbling the location where this is violated.
For each game we can also use initial configurations of pebbled tuples instead of starting from empty configurations.
Remark: Color refinement and pebble games are not necessarily restricted to finite groups. While not clear that the results are computable, they still may be of theoretical interest. The same goes for the logics defined next.
3.3 Logics with counting
As for graphs, the dimensional refinement on groups can also be interpreted in terms of firstorder counting logic.
Recall the central aspects of first order logic. There is a countable set of variables . Formulas are inductively defined so that is a formula for all pairs of variables and if is a formula then , ,, and are formulas. The semantics are defined in the obvious way. First order logic with counting allows additionally formulas of the form with the semantic meaning that there are at least distinct elements that satisfy .
To define logics on groups we need to additionally define a relation that relates to the group multiplication.
Version I:
In Version I we add a ternary relation with which we can create terms of the form . The semantic interpretation is that holds if . We call the first order logic with counting on groups arising this way and let be its variable fragment.
Version II:
For we use a different relation to access multiplication: The relation holds, where is a word in the , if multiplying the elements according to gives the trivial element. For example in an abelian group the relation would hold for all elements . The relation would only hold if is the trivial element. We let be the variable fragment of the logic. Note that for it actually suffices to use only entries in the relation.
Version III:
The natural choice of logic for Version III is of course the classical first order logic with counting on graphs as discussed in the preliminaries (Subsection 2.3), where we have the relation to encode edges. For notational consistency we define to be the variable fragment of this logic.
3.4 Equivalence between the different concepts
For each of the versions we have defined, we sketch the arguments for equivalence of the expressive power between the WLalgorithm, the pebble game, and the corresponding logic. Let us fix groups and of the same order. The argument basically follows other well known arguments to show such equivalences (see e.g., [7]).
Theorem 3.2.
Two groups and are distinguished by the WLrefinement (Version ) if and only if the same holds for the bijective pebble game (Version ).
Remark: Let us remark on a small detail where the group situation can differ from that of graphs. Note that in our definition of the game, the winning condition is only ever checked after Step 1. We could also check the winning condition when pebble pairs are situated on the group after a round is finished. For this we would need an initial coloring that works with tuples. For graphs this change does not make a difference, since the winning condition only ever depends on 2 pebble pairs. Similarly for Version I, where the winning condition occurs due to 3 pebble pairs, if then it is irrelevant when we check the winning condition. However, for Version II we are not so sure how the power of the game changes, when altering the moment at which the winning condition is checked.
Theorem 3.3.
if and only if there is a sentence in that holds on one of the groups but not the other.
Lemma 3.4.
Suppose and . If and obtain different colors in the th iteration of dimensional WLrefinement then Spoiler can win the pebble game in moves on initial configuration . (Here we use the same version for WLrefinement and pebble game.)
Proof Sketch.
(Version I.) For there is nothing to show. Assume now that . By assumption and are different which means that either we already have or there is no colorpreserving matching between the tuples for and tuples for . In other words, no matter which bijection with Duplicator chooses there will be some and some position such that and Spoiler can make progress by changing the th pebble from to . By induction, Spoiler now has a winning strategy with moves while having moved at most once.
(Version II.) If then and differ with respect to marked isomorphism, thus Spoiler can win without moving at all. For the argument is the same as before since the refinement steps are defined equally.
Lemma 3.5.
Suppose and . If Spoiler can win the pebble game in moves on initial configuration then and obtain different colors in the th iteration of dimensional WLrefinement. (Again we use the same version for WLrefinement and pebble game.)
Proof sketch.
(Version I.) If then the initial configuration is already a winning one for Spoiler which is by definition the same as tuples getting different initial colors. By induction, for any bijection Duplicator may choose, Spoiler can reach in one move a configuration where and for some position and such that . Since this is true for any possible bijection, the tuples and already have to differ with respect to .
(Version II.) The argument is the same as for Version I.
It remains to argue the equivalences between the logic and the pebble game for each of the versions. This again basically follows from known techniques.
We first argue that for Version I and II the quantifier free formulas of the variable fragment of each version characterize the initial colorings.
Lemma 3.6.
There is a quantifier free variable formula distinguishing tuples and if and only if these tuples differ in their initial coloring in version .
Proof.
If then distinguishes the tuples and if and only if there is an atomic statement of the form or , interpreted as , with respect to which and differ. This is precisely the definition of the Version I initial coloring. For Version , if some word over is (non)trivial but the corresponding word over is not then clearly mapping to does not extend to an isomorphism. Assume now that and have different marked isomorphism types and w.l.o.g. we have . By assumption every injective map between these groups extending is not multiplicative. This fact can be expressed in terms of a suitable word over symbols separating from . ∎
Lemma 3.7.
Suppose and . For each Version , the tuples and are distinguished by WL if and only if there is a formula in such that .
Proof sketch.
(Version I.) For Version I, due to Lemma 3.6, the expressive power of the initial coloring is precisely the expressive power of quantifierfree formulas. The distinguishing power of the WLalgorithm on groups is thus equal to distinguishing power of the classical algorithm executed on a structure that is already endowed with the initial coloring. The equivalence between the variable fragment of the logic and the WL algorithm for Version I on groups thus follows from the respective equivalence for graphs shown in [7].
(Version II.) For Version II the argument is the same as for Version I except that we add the following observation: Since and are finite groups there is only a finite number of nonequivalent quantifier free formulas over . By Lemma 3.6 tuples can be distinguished exactly if they obtain different colors in the initial coloring of Version II.
(Version III.) This is again a classical result [7]. ∎
3.5 Relationship between the different WLalgorithm versions
Next, we want to relate different versions to each other and we will do so by exploiting the equivalence to pebble games.
Definition 3.8.
Consider the pebble game on graphs and and assume that a pair of pebbles is placed on vertices corresponding to multiplication gadgets and (but not on vertices corresponding to group elements). Then the pairs and will be called implicitly pebbled. Note that implicit pebbles always induce a pairing of group elements.
Intuitively, pebbling a vertex in a multiplication gadget is as strong as pebbling two group elements simultaneously, hence the definition of implicit pebbles. It can be shown that Duplicator has to respect the gadget structure and in particular the multiplication structure of the implicitly pebbled elements.
Theorem 3.9.
For all we have .
The rest of this section spans the proof of this theorem.
Lemma 3.10.
Consider the pebble game on graphs and . If and one of the following happens

Duplicator chooses a bijection with ,

after choosing a bijection, there is a pebble pair , for which pebble is on some vertex of (not on , or ) and is on some vertex of but , or

the map induced on group elements pebbled or implicitly pebbled by pebbles does not extend to a group isomorphism between the corresponding generated subgroups
then Spoiler can win the game.
Proof.

In vertices corresponding to group elements have degree while other vertices have degree or .

From now on always assume . Write and . Let for some and put a pebble other than on and the corresponding one on . If pebble is not on the same type of vertex (i.e., Type , , or , see Figure 2) as pebble then Spoiler wins, since either the vertices have different degrees or their neighbors have different degrees.
Now the pebbled vertex in is connected to directly or via a path of nongroup element vertices and for and either this is not the case or the path uses different types of vertices. Using a third pebble pair Spoiler can explore this path and win on a configuration of three pebbles.

By assumption there are at most implicitely pebbled pairs of group elements corresponding to at most pebbles that are currently on the board. We may assume that there are exactly such pairs, say. By the second part of this lemma Duplicator has to choose some bijection such that respecting the pairing induced by indirectly pebbled group elements. By assumption this correspondence does not extend to an isomorphism between and so there must be a smallest word over such that
Spoiler can use an additional pair of pebbles to fix this image of . Now Duplicator chooses a new bijection on the remaining group elements. There is now either a smaller word over with this property, in which case Spoiler moves the last pebble pair we just introduced to this new word and its image, or is still minimal with this property. The first case can only occur finitely many times and if Spoiler wins by part . Thus assume that is still minimal. But then
and using a second additional pair of pebbles (now a total of at most pairs of actual pebbles) Spoiler can also fix the image of to be and clearly wins from this configuration in at most two further rounds.
∎
Lemma 3.11.
If then .
Proof.
Assume that Spoiler wins the pebble game in Version II. The idea is to simultaneously play a Version II game on groups and a Version III game on graphs. For this purpose we have to be able to compare pebbleconfigurations from the different games. Let be a configuration of pebbled tuples on and and call a configuration on graphs admissible if it looks as follows: There is one pair of pebbles on the multiplication gadgets and , another pair on and and so forth. If
is odd then there is another pair of pebbles on vertices
and and there are no other pebbles on the graphs. Note that the number of pebbles on each graph is and that implicit pebbles (together with the pebbles on ) correspond exactly to the pebbles on groups in Version II. Using Lemma 3.10 we can assume throughout the game that Duplicator chooses bijections on graphs that restrict to bijections on groups and that those restrictions respect implicit pebbles or otherwise Spoiler would win Version III right away. That means that given an admissible configuration, the bijection Duplicator chooses in Version III can be used as a Version II bijection as well. Spoiler will then move in Version II and we argue that Spoiler can win in Version III or force Duplicator into another admissible configuration. Since Spoiler can choose arbitrary moves in Version II, by assumption Spoiler will win in Version II at some point. Using Lemma 3.10 again, we see that Duplicator will eventually lose the Version III game on this configuration. It remains to argue that Spoiler can maintain admissible configurations. Assume Duplicator chose a bijection as above. There are two cases: Spoiler moves a pebble or introduces a new one. Suppose Spoiler introduces a new pebble pair on and in the groups. If is even the new pebble on is grouped with the already existing pebble on . In the graph, Spoiler will put a pebble on . The corresponding pebble should be put on to obtain an admissible configuration. But since was not necessarily (implicitly) pebbled it may be the case that does not map to . To fix this, Spoiler first pebbles and directly, using one additional pebble, and asks for another bijection. By Lemma 3.10 this bijection must now map to and Spoiler reaches an admissible configuration in Version III in two more moves, removing the additional pebble again. The case where a pebble is moved rather than newly introduced can be treated in the same way. ∎Lemma 3.12.
If then .
Proof.
We now want to reverse the argument from the last lemma. Given a pebbleconfiguration on graphs and we call a pebbleconfiguration on groups admissible if the following holds: for each pair of pebbles on elementvertices the corresponding elements in and are pebbled as well and for each pair of pebbles on nonelement vertices, that is, pebbles on gadgets and , there are pairs of pebbles on and , and , respectively. Note that, w.l.o.g., (non)elementvertices are only pebbled among each other by Lemma 3.10. Also the number of pebbles on groups in an admissible configuration is at most twice the number of pebbles on graphs. Since Spoiler can move two implicit pebbles at once in Version III we will look at two consecutive rounds of the Version I game at once. Let Duplicator choose a bijection in the Version I game. For each possible Spoiler move introducing an additional pebble on , Duplicator has to commit to one bijection on the new configuration. We can force Duplicator to choose this bijection in the corresponding configuration from now on without changing the deterministic outcome of the game, because Duplicator is allowed to choose it freely once. This gives rise to a bijection between pairs of group elements mapping to . Note that this happens without actually making moves, rather think of Duplicators strategy as being precomputable by Spoiler due to the deterministic nature of the game. The map on pairs can now be interpreted as a mapping between element vertices together with a mapping of corresponding multiplication gadgets and will be used as the next Duplicator move in the Version III game. If the Spoiler move in Version III moves two implicit pebbles at once, Spoiler can reach an admissible configuration in three rounds (one additional round for discarding the additional pebble) in the Version I game while Duplicator chooses bijections according to the precomputed strategy.
Finally, before Spoiler wins in Version III, Spoiler will win in Version I. More precisely, as long as the map on pebbles in the Version I game is multiplicative, the corresponding map induced on the pebbled subgraph will be a graph isomorphism, since multiplicativity on pebbles can be expressed equivalently in terms of mapping gadgets accordingly. ∎
Proof of Theorem 3.9.
The first inclusion is clear. The other inclusions are the content of the previous lemmas. ∎
We remark that the additive constants in the Theorem 3.9 could be improved for by reusing pebbles, but we do not worry about explicit constants at this point. It is also possible to show directly.
4 Embedding graphs into finite groups
Next, we describe a construction of finite groups from graphs such that structural properties of the resulting groups are primarily determined by the graphs. We will make this statement more precise in the following. From now on fix an odd prime .
Definition 4.1.
For each natural number there is a relatively free group of exponent and (nilpotency) class generated by elements. It admits a finite presentation
where consists of the following relations:

For all there is a relation , and

for all there is a relation .
Thus, the group is generated by , each of these generators is an element of order , and the commutator of two generators commutes with every generator and thus every element of the group. It follows from these properties that elements of can be uniquely written as
where exponents are defined modulo . In particular, .
The main goal is to construct quotients of using graphs on vertex set as templates in a way that translates combinatorial similarity of the graphs (with respect to WeisfeilerLemanrefinement) to similar subgroup profiles. We will see that this affects other isomorphism invariants as well.
Definition 4.2.
To each (simple, undirected) graph and prime number we assign a finite exponent group of nilpotency class via
Thus, in two generators commute, if the corresponding vertices form an edge in . We usually identify with and use the latter to refer to the vertex as well as the respective element of . We fix an order on generators and call these the standard generators for . The particular presentation above is called the presentation of from .
It turns out that this construction has also been used in other contexts. It is sometimes called Mekler’s construction in the literature (see [27] for Mekler’s original work) and has been primarily investigated for infinite graphs with respect to model theoretic properties. We first collect some possibly well known combinatorial and group theoretic properties.
Lemma 4.3.
We have and the vertices of form a generating set of of minimal cardinality.
Proof.
By construction has exponent and thus (since for groups the Frattinisubgroup is the minimal subgroup with elementary abelian quotient). The cardinality of a minimal generating set of is the dimension of the space which is now equal to . We have
showing the claim. ∎
Lemma 4.4.
Denote by the number of nonedges in . Then , i.e., the set of nonedges of forms a basis in .
Proof.
We have for some normal subgroup with and since commutators are central in we have where . ∎
This also gives us normal forms for elements of .
Corollary 4.5.
Let be a (simple) graph. Then we have . In particular, every element of can be written in the form
where is the set of nontrivial commutators between generators (i.e., the nonedges of the graph ) and each is uniquely determined modulo .
We will see that a lot of information on commutation and centralizers can be deduced from directly. We first need to recall some well known properties of commutators in (nilpotent) groups.
Lemma 4.6 (Commutator relations).
Let be a group of nilpotency class . Then for all we have

and

.
In particular for all we have .
Proof.
Recall that nilpotency class means that all commutators are central in . We thus have and we have that
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