aa r X i v : . [ m a t h . G R ] M a r Graphs defined on groups
Peter J. CameronUniversity of St Andrews [email protected]
Abstract
These notes concern aspects of various graphs whose vertex set is agroup G and whose edges reflect group structure in some way (so that,in particular, they are invariant under the action of the automorphismgroup of G ). The particular graphs I will chiefly discuss are the powergraph, enhanced power graph, deep commuting graph, commutinggraph, and non-generating graph.My main concern is not with properties of these graphs individ-ually, but rather with comparisons between them. The graphs men-tioned, together with the null and complete graphs, form a hierarchy(as long as G is non-abelian), in the sense that the edge set of any oneis contained in that of the next; interesting questions involve when twographs in the hierarchy are equal, or what properties the difference be-tween them has. I also consider various properties such as universalityand forbidden subgraphs, comparing how these properties play out inthe different graphs. (There are so many papers even on the powergraph that a complete survey is scarcely possible.)I have also included some results on intersection graphs of sub-groups of various types, which are often in a “dual” relation to one ofthe other graphs considered. Another actor is the Gruenberg–Kegelgraph, or prime graph, of a group: this very small graph has a sur-prising influence over various graphs defined on the group. I say littleabout Cayley graphs, since (except in special cases) these are not in-variant under the automorphism group of G .Other graphs which have been proposed, such as the nilpotence,solvability, and Engel graphs, will be touched on rather more briefly.My emphasis is on finite groups but there is a short section on resultsfor infinite groups. roofs, or proof sketches, of known results have been includedwhere possible. Also, many open questions are stated, in the hope ofstimulating further investigation.The graphs I chiefly discuss all have the property that they contain twins , pairs of vertices with the same neighbours (save possibly oneanother). Being equal or twins is an equivalence relation, and theautomorphism group of the graph has a normal subgroup inducingthe symmetric group on each equivalence class. For some purposes, wecan merge twin vertices and get a smaller graph. Continuing until nofurther twins occur, the result is unique independent of the reduction,and is the trivial 1-vertex graph if and only if the original graph is a cograph . So I devote a section to cographs and twin reduction, andanother to the consequences for automorphism groups. In addition,I discuss the question of deciding, for each type of graph, for whichgroups is it a cograph. Even for the groups PSL(2 , q ), this leads todifficult number-theoretic questions.There are briefer discussions of general Aut( G )-invariant graphs,and structures other than groups (such as semigroups and rings). Contents
10 Automorphisms 4811 Above and beyond the hierarchy 50
12 Intersection graphs 55
14 Infinite groups 61
15 Beyond groups 64
There are a number of graphs whose vertex set is a group G and whose edgesreflect the structure of G in some way, so that the automorphism group of G acts as automorphisms of the graph. These include the commuting graph(first studied in 1955), the generating graph (from 1996), the power graph(from 2000), and the enhanced power graph (from 2007), all of which havea considerable and growing literature. A relative newcomer, not publishedyet, is the deep commuting graph.This paper does not aim to be a survey of all these areas, which wouldbe far too ambitious a task. Rather, I am interested in comparisons amongthe different graphs. In particular, there is a hierarchy containing the nullgraph, power graph, enhanced power graph, deep commuting graph, commut-ing graph, non-generating graph (if the group is non-abelian), and completegraph: the edge set of each is contained in that of the next.These graphs have some similarities: for example, the enhanced powergraphs, commuting graphs, deep commuting graphs, and generating graphsof finite groups all form universal families (that is, every finite graph is em-beddable in one of these graphs for some group G ). However, the proofs ofthis require rather different techniques for the different graphs.Another question, about which relatively little is currently known, con-cerns the differences between graphs in the hierarchy. Even rather basicquestions such as connectedness are unstudied for most of these, althoughSaul Freedman and coauthors have results on the difference between thenon-generating graph and the commuting graph (and, at top and bottom,4he difference between the complete graph and the non-generating graph isthe generating graph, while the difference between the power graph and thenull graph is the power graph, both of which have an extensive literature).For some of these graphs, either the complementary graph was definedfirst, or the graph and its complement were studied independently. For ex-ample, the generating graph of a finite group preceded the non-generatinggraph; and Neumann’s theorem, that if the non-commuting graph of an infi-nite group contains no infinite clique then it has finite clique number, clearlyhas an equivalent formulation in terms of the coclique number of the commut-ing graph. Also, several of the questions I consider have easy translations forthe complement graph; for example, the universality results just mentionedimmediately show that the complementary graphs are universal too. Twinreduction and the related concept of cograph do not distinguish between agraph and its complement. I have chosen to focus on those graphs whichform a hierarchy; if the complements were preferred, the hierarchy wouldreverse.A curious feature is the appearence of the Gruenberg–Kegel graph, whichdetermines (or almost determines) various features of the commuting graphand the power graph. The vertex set of this graph is not the group, but themuch smaller set of prime divisors of the group order. For example, if G hastrivial centre, its reduced commuting graph (with the identity removed) isconnected if and only if its Gruenberg–Kegel graph is. Conversely, for allgraph types in the hierarchy except possibly the non-generating graph, thecorresponding graph on G determines the Gruenberg–Kegel graph of G .Authors who have studied these have used a variety of notations for them.I have tried to use a consistent and helpful notation, for example, Pow( G )and Com( G ) for the power graph and commuting graph, respectively, of G .The final, brief sections concern more general graphs defined on groupsand invariant under group automorphisms; the graphs of the hierarchy on in-finite groups; and extensions to other algebraic structures such as semigroupsand rings.Since much is not known, I have tried to emphasise open problems through-out.Computations reported here were performed using GAP [51], with thepackages GRAPE [97] for handling graphs, HAP [47] for computing Schurand Bogomolov multipliers, and LOOPS [84] for Moufang loops. Generatorsfor specific groups were taken from the On-Line Atlas of Finite Groups [105].I’m grateful to several people, especially Alireza Abdollahi, Saul Freed-5an, Michael Giudici, Michael Kinyon, Bojan Kuzma and Natalia Maslova,for helpful comments on a previous version.
I will denote a typical graph by Γ, with vertex set V (Γ) and edge set E (Γ).If A is a subset of V (Γ), then the induced subgraph of Γ on A is the graphwith vertex set A whose edges are those of Γ contained in A . A completegraph is one in which all pairs of vertices are joined; a null graph is one withno edges. A clique (resp. coclique ) is a set of vertices on which the inducedsubgraph is complete (resp. null).For a graph Γ, • the clique number ω (Γ) is the size of the largest clique; • the clique cover number θ (Γ) is the minimum number of cliques whoseunion is the vertex set of Γ; • the independence number or coclique number α (Γ) is the size of thelargest coclique; • the chromatic number χ (Γ) is the smallest number of independent setswhose union is the vertex set of Γ (so-called because it is the smallestnumber of colours needed to colour the vertices so that adjacent verticesget different colours).Note that the independence number and clique cover number of Γ are equalto the clique number and chromatic number of the complement of Γ (thegraph with the same vertex set, whose edges are the non-edges of Γ).It is clear that ω (Γ) χ (Γ) and α (Γ) θ (Γ). The graph Γ is called perfect if every induced subgraph has clique number equal to chromatic number.The Weak Perfect Graph Theorem of Lov´asz [78] asserts that, if Γ is perfect,then so is its complement; the
Strong Perfect Graph Theorem of Chudnovsky et al. [41] asserts that Γ is perfect if and only if it does not contain a cycle ofodd length greater than 3 or the complement of one as an induced subgraph.The comparability graph of a partial order P = ( A, ) is the graph withvertex set A , in which a and b are joined if either a b or b a . Dilworth’sTheorem [45] asserts that the comparability graph of a partial order and itscomplement are both perfect. (The first part is easy, the second less so butfollows from the first using the Weak Perfect Graph Theorem.)6roups will almost always be finite here; my notation for finite groups isstandard. (The reason for calling a graph Γ is to avoid conflict of notation,since G will be a typical group.) One topic I will not consider, except in this section, concerns Cayley graphs.A
Cayley graph for the group G is a graph on the vertex set G which isinvariant under right translation by elements of G . (Some authors use lefttranslation; the two concepts are equivalent, and the inversion map on G converts one into the other.) Equivalently, if S is an inverse-closed subset of G \ { } , then the Cayley graph Cay( G, S ) is the graph with vertex set G inwhich g and h are adjacent whenever gh − ∈ S .One reason for not considering these is that they have a huge literature,far more than I can survey here. I have heard the view expressed thatalgebraic graph theory is the study of Cayley graphs of finite groups (in fact,it is broader than this, but Cayley graphs are an important topic); while it iscertainly arguable that geometric group theory is the study of Cayley graphsof finitely generated infinite groups.The other is that Cayley graphs are not in general preserved by the au-tomorphism group of G . I will say a few words about this.Suppose that the set S is a normal subset of G , that is, closed under con-jugation. Then Cay( G, S ) is invariant under both left and right translation.Such a graph is sometimes called a normal Cayley graph , see for example[66, 75]. However, the reader is warned that more recently this term hasbeen used in a completely different sense: a Cayley graph Γ = Cay(
G, S ) isnormal if the group of right translations of Γ is a normal subgroup of Aut(Γ),see for example [107].To avoid confusion, I propose to call a normal Cayley graph in the firstsense above an inner-automorphic
Cayley graph. Note that this conditionis equivalent to saying that the graph is invariant under both left and righttranslation. (For the composition of right translation by g and left translationby g − is conjugation by g , and so a graph invariant under two of these mapsis invariant under the third also.) Proposition 1.1
The Cayley graph
Cay(
G, S ) is inner-automorphic if andonly if S is a union of conjugacy classes in G . G are the relations of the conjugacy class association scheme on G , see [30,54, 106] (the last of these references calls this structure the group scheme of G ).I shall call the Cayley graph Cay( G, S ) automorphic if it is invariant underthe whole of Aut( G ), that is, if S is a union of orbits of Aut( G ) acting on G .(Thus, if Cay( G, S ) is automorphic, then its automorphism group containsthe holomorph of G .) These graphs would fall under the rubric consideredhere, although I shall not be discussing them further. This section introduces the specific graphs on a group that I will be mainlyconcerned with.
Let G be a finite group. The commuting graph of G is the graph with vertexset G in which two vertices x and y are joined if xy = yx . This graph wasintroduced by Brauer and Fowler in their seminal paper [26] showing thatonly finitely many groups of even order can have a prescribed centraliser.(Brauer and Fowler do not use the word “graph”, but define the graph metricin the induced subgraph of the commuting graph on G \ { } and use this.The argument begins by showing that, if there are at least two conjugacyclasses of involutions, then any two involutions have distance at most 3 by apath in the commuting graph which avoids the identity.)The commuting graph has had further applications in group theory. Ver-tices in Z ( G ) are joined to everything, and for investigating questions suchas connectedness these are often removed; this makes no difference here.Also the definition puts a loop at every vertex. There is good reason fordoing this. It follows from results of Jerrum [68] on the “Burnside process”that the limiting distribution of the random walk on the commuting graphwith loops is uniform on conjugacy classes – that is, the limiting probabilityof being at a vertex is inversely proportional to the size of its conjugacyclass. This is useful in finding representatives of very small conjugacy classesin large groups. 8losely related is the fact that the commuting ratio of a group G , theprobability that two randomly-chosen elements commute, is the ratio of thenumber of ordered edges of the commuting graph (including loops) to | G | :see for example [58, 46].But for my purposes here, I will imagine that the loops have been silentlyremoved.As with all of these graphs, we can ask: Which groups are characterisedby their commuting graphs? For example, abelian groups of the same orderhave isomorphic commuting graphs, as do the dihedral and quaternion groupsof order 8. Figure 1 the commuting graphs of the two groups D = h a, b : a = 1 , b = 1 , b − ab = a − i and Q = h a, b : a = 1 , b = a , b − ab = a − i . ✉✉ ✉✉ (cid:0)(cid:0)(cid:0)❅❅❅❭❭❭❭❭❭ ✉✉ ▲▲▲▲▲▲✜✜✜✜✜✜ ✉✉ ☞☞☞☞☞☞ a a a ba baba b Figure 1: Commuting graph of D or Q It was conjectured in [4] and proved in [7, 60, 98] that any non-abelianfinite simple group is characterised by its commuting graph.I note in passing an application of the commuting graphs of finite groupsto the structure of finite quotients of the multiplicative group of a finite-dimensional division algebra by Segev [93].To conclude this section, a simple observation about the commutinggraph:
Proposition 2.1
A maximal clique in the commuting graph of G is a max-imal abelian subgroup of G . The deep commuting graph of a finite group G was introduced very re-cently [36]. Two elements of G are joined in the deep commuting graphif and only if their preimages in every central extension of G (that is, every9roup H with a central subgroup Z such that H/Z ∼ = G ) commute. Morespecifically, take the commuting graph of a Schur cover [92] of G (this is acentral extension H of largest order such that Z is contained in the derivedgroup of H ), and take the induced subgraph of the commuting graph of H ona transversal to Z . It can be shown that the resulting graph is independentof the choice of Schur cover.For example, D and Q are Schur covers of the Klein group V ; Figure 1shows that the deep commuting graph of the Klein group is the star K , ,though its commuting graph is the complete graph K .We note in particular that the deep commuting graph is equal to thecommuting graph if the Schur multiplier of G (the central subgroup Z in aSchur cover) is trivial. The converse is false, as we will see in Proposition 3.3. Question 1
Is it true that a non-abelian finite simple group is characterisedby its deep commuting graph?
The directed power graph of G is the directed graph with vertex set G , withan arc x → y if y = x m for some integer m . The power graph of G is the graphobtained by ignoring directions and double arcs; in other words, x is joined to y if one of x and y is a power of the other. It is clearly a spanning subgraphof the commuting graph. The power graph was introduced by Kelarev andQuinn [71].The directed power graph is a partial preorder , that is, a reflexive andtransitive relation on G ; and the power graph is its comparability graph (twovertices joined if and only if they are related in the preorder). Comparabilitygraphs of partial preorders and partial orders form the same class. For clearlyevery partial order is a partial preorder. For the converse, if ( A, ) is a partialpreorder, then the relation ≡ defined by a ≡ b if a b a is an equivalencerelation; putting a total order on each equivalence class gives a partial orderwith the same comparability graph as ( A, ). It follows from Dilworth’stheorem that the power graph of a finite group is perfect. This was firstproved by Feng et al. [48]; see also [1].The power graph does not uniquely determine the directed power graph;for example, if G is the cyclic group of order 6, then the identity and thetwo generators are indistinguishable in the power graph (they are joined to10ll other vertices), but one is a sink and the other two are sources in thedirected power graph. However, the following is shown in [31]: Theorem 2.2
If two finite groups have isomorphic power graphs, then theyhave isomorphic directed power graphs.
This is false for infinite groups; see [33].For a more extensive survey of power graphs, I refer to [2].
In 2007, Abdollahi and Hassanabadi [5, 6] studied a graph they called the noncyclic graph of a group G , in which two vertices x and y are joined if h x, y i is not cyclic. (For technical reasons they excluded the set of isolatedvertices, the so-called cyclicizer of G .) Later in this paper, some of theirresults will be discussed.For comparison with the other graphs considered in this paper, I willtake the complement of their graph, which was independently defined in thepaper [1] under the name enhanced power graph of G . Also, I will not initiallyassume that vertices joined to all others are excluded; we will examine suchvertices later.Thus, the enhanced power graph of a group G has vertex set G , with x and y joined if and only if h x, y i is cyclic. Equivalently, x and y are joinedif there is an element z ∈ G such that each of x and y is a power of z .This graph was introduced to interpolate between the power graph and thecommuting graph, but has now been studied in its own right, especially bySamir Zahirovi´c and coauthors [108, 109], and is in some respects easier tohandle than the power graph.The enhanced power graph can be obtained from the directed power graphby joining two vertices if both lie in the closed out-neighbourhood of somevertex. Thus, if two groups have isomorphic power graphs, then they haveisomorphic enhanced power graphs. The converse is also true, see [109]: Theorem 2.3
For a pair of finite groups, the following are equivalent:(a) the power graphs are isomorphic;(b) the directed power graphs are isomorphic;(c) the enhanced power graphs are isomorphic.
11t is, however, not true that the power graph, directed power graph, andenhanced power graph of G have the same automorphism group. For G = C ,all three automorphism groups are different. Question 2
Is there a simple algorithm for constructing the directed powergraph or the enhanced power graph from the power graph, or the directedpower graph from the enhanced power graph?Note that the enhanced power graph of G is a union of complete subgraphson the maximal cyclic subgroups of G . Similarly, the commuting graph isa union of complete subgraphs on the maximal abelian subgroups. In fact,more is true: Proposition 2.4
A maximal clique in the enhanced power graph of G is amaximal cyclic subgroup of G . This follows from the fact that if a set of elements of G have the propertythat any two generate a cyclic subgroup, then the whole set generates a cyclicsubgroup: see [1, Lemma 32].At this point I mention another graph which I will not discuss in somuch detail. It has been studied under the name intersection graph (see forexample [40, 62]), but I wish to reserve this term for a different concept.Since, as we will see, it is dual (in a certain vague sense) to the enhancedpower graph, I will call it the dual enhanced power graph of G , and denote itDEP( G ). Since the identity is always isolated in this graph (unlike the othergraphs discussed so far), it is natural to remove it and define the vertex setof DEP( G ) to be G \ { } .Two non-identity elements x and y are joined in the dual enhanced powergraph of G if h x i ∩ h y i is not the identity. Recall that two vertices x, y arejoined in the enhanced power graph if they have a common in-neighbour inthe directed power graph; we see that two vertices x and y are joined inDEP( G ) if they have a common out-neighbour different from the identity.In particular, we see that the directed power graph determines the dualenhanced power graph. But, unlike the case with the power graph and en-hanced graph, it does not work in the reverse direction. If G and H are thecyclic group and quaternion group of order 8, then DEP( G ) and DEP( H )are complete graphs on 7 vertices, but the directed power graphs of thesetwo groups are not isomorphic. 12 .5 The generating graph The generating graph of a finite group G has vertex set G , with x and y joined if and only if h x, y i = G . If the minimum number of generators of G is greater than 2, then the generating graph is the null graph. If G is cyclic,then its generating graph has loops; we will not be too much interested inthis case. Note that, by the Classification of Finite Simple Groups, everynon-abelian finite simple group is 2-generated.The generating graph was introduced in [76], and studied further in [27].The results are often phrased in terms of the spread , a graph-theoretic pa-rameter defined as follows: Γ has spread (at least) k if every set of k verticeshas a common neighbour. Thus a graph has spread 1 if it has no isolatedvertex, while spread 2 is stronger than having diameter at most 2.In [27] it was shown that the generating graph of a non-abelian finitesimple group has positive spread . A substantial strengthening has recentlybeen proved by Burness et al. [29]: Theorem 2.5
For a finite group G with reduced generating graph Γ , thefollowing three conditions are equivalent:(a) any non-identity vertex has a neighbour in the generating graph (thatis, Γ has spread );(b) any two non-identity vertices have a common neighbour in the gener-ating graph (that is, Γ has spread );(c) any proper quotient of G is cyclic. In particular, the conditions hold for a non-abelian finite simple group.However, it is false for infinite groups: even (a) can fail for 2-generator infinitegroups with all proper quotients cyclic [43].In the sequel, I will often consider the non-generating graph , the comple-ment of the generating graph.
These graphs are given ad hoc names in the literature, but since I will betalking about all of them here, I prefer to give them names which help todistinguish them. Thus, the commuting graph of G will be Com( G ); thedeep commuting graph DCom( G ); the power graph Pow( G ); the directed13ower graph DPow( G ); the enhanced power graph EPow( G ); the generatinggraph Gen( G ); and the non-generating graph NGen( G ). In each case, thevertex set is the group G . Sometimes I refer to the reduced graph of one ofthe above types, and denote it by a superscript − ; this usually means thatthe identity element is deleted from the vertex set.There are inclusions between these graphs, as follows. Here E (Γ) denotesthe edge set of a graph Γ; thus E (Γ ) ⊆ E (Γ ) means that Γ is a spanningsubgraph of Γ (a subgraph using all of the vertices and some of the edges). Proposition 2.6
Let G be a finite group.(a) E (Pow( G )) ⊆ E (EPow( G )) ⊆ E (DCom( G )) ⊆ E (Com( G )) .(b) If G is non-abelian or not 2-generated, then E (Com( G ) ⊆ E (NGen( G )) . Proof (a) All is obvious except possibly the inclusion of E (EPow( G )) in E (DCom( G )). So suppose that h x, y i is a cyclic subgroup of G , and let H bea central extension of G , with H/Z ∼ = G . The lift of h x, y i is the extensionof a central subgroup Z by a cyclic group, and hence is abelian; so the liftsof x and y commute in H .(b) If G is not 2-generated, then NGen( G ) is the complete graph, and theresult is clear. If G is non-abelian, it cannot be generated by two commutingelements. (cid:3) Because of this, I will refer to the null graph, power graph, enhancedpower graph, deep commuting graph, commuting graph, non-generating graph(in the case that G is non-abelian) and complete graph on G as the graphhierarchy , or just hierarchy , of G .To conclude this section, I note that the dual enhanced power graph doesnot fit very well into the hierarchy. Recall that a superscript − means thatthe identity is removed; the identity is isolated in DEP( G ) so it is natural toremove it from this graph also. Proposition 2.7 (a) For any finite group G , E (Pow − ( G )) ⊆ E (DEP( G )) .(b) If Z ( G ) = 1 , then E (DEP( G )) ⊆ E (NGen − ( G )) .(c) In general DEP( G ) is incomparable with the other graphs in the hier-archy. roof (a) Suppose that { x, y } is an edge of Pow( G ) with x, y = 1. Withoutloss of generality, x is a power of y ; then h x i ∩ h y i = h x i .(b) Suppose that Z ( G ) = 1, and that { x, y } is an edge of DEP( G ). Let h x i ∩ h y i = h z i , with z = 1. Then x, y ∈ C G ( z ) < G , so h x, y i 6 = G .(c) If G = C , then EPow − ( G ), and all higher graphs in the hierarchy,are complete, but DEP( G ) is not. If G = Q , then Com − ( G ), and all lowergraphs in the hierarchy, are incomplete, but DEP( G ) is complete. (cid:3) Question 3
For which groups G , and for which types X of graph in thehierarchy, does DEP( G ) = X − ( G ) hold? A related graph will play a role in the investigation in several places. The
Gruenberg–Kegel graph , also known as the prime graph , of a finite group G has vertex set the set of prime divisors of the order of G ; vertices p and q arejoined by an edge if and only if G contains an element of order pq .The graph was introduced in an unpublished manuscript by Gruenbergand Kegel to study the integral group ring of a finite group, and in particularthe decomposability of the augmentation ideal: see [57]. The main structuralresult was published by Williams (a student of Gruenberg) in [104]. It as-serts that groups whose Gruenberg–Kegel graph is disconnected have a veryrestricted structure. Theorem 2.8
Let G be a finite group whose Gruenberg–Kegel graph is dis-connected. Then one of the following holds:(a) G is a Frobenius or -Frobenius group;(b) G is an extension of a nilpotent π -group by a simple group by a π -group,where π is the set of primes in the connected component containing . A 2 -Frobenius group is a group G with normal subgroups H and K with H ≤ K such that • K is a Frobenius group with Frobenius kernel H ; • G/H is a Frobenius group with Frobenius kernel
K/H .15 typical example is the group G = S , with K = A , H = V (the Kleingroup), and G/K ∼ = S .Williams went on to examine the known finite simple groups to determinewhich ones could occur in conclusion (b) of the Theorem. He could not handlethe groups of Lie type in characteristic 2; this was completed by Kondrat’evin 1989, and some errors corrected by Kondrat’ev and Mazurov in 2000.The next result indicates that the Gruenberg–Kegel graph is closely con-nected with our hierarchy of graphs. Theorem 2.9
Let G and G be groups whose power graphs, or enhancedpower graphs, or deep commuting graphs, or commuting graphs, are isomor-phic. Then the Gruenberg–Kegel graphs of G and G are equal. Proof
The four possible hypotheses each imply that G and G have thesame order, so their GK graphs have the same set of vertices.We show that in all cases except the power graph, primes p and q areadjacent in the GK graph of G if and only if there is a maximal cliquein the graph on G with size divisible by pq . This is clear in the cases ofthe enhanced power graph and the commuting graph; for, as we observedearlier, the maximal cliques in these are maximal cyclic subgroups or maximalabelian subgroups of G respectively (Propositions 2.1 and 2.4), and if theirorder is divisible by pq then they contain elements of order pq . Conversely anelement of order pq is contained in a maximal cyclic (or abelian) subgroup.Consider the deep commuting graph of a group G . Let H be a Schurcover of G , with H/Z ∼ = G . A maximal clique has the form A = B/Z , where B is a maximal abelian subgroup of H (containing Z ). So A is an abeliansubgroup of G , and if pq divides | A | then A contains an element of order pq . Conversely, suppose that p and q are joined in the GK graph, and let x and y be commuting elements of orders p and q in G , and a and b theirlifts in H . Then a and b are contained in h Z, ab i , which is an extension of acentral subgroup by a cyclic group and hence is abelian; so a and b commute.Choosing a maximal abelian subgroup of H containing a and b and projectingonto G gives a maximal clique in DCom( G ) with order divisible by pq .This fails for the power graph. Instead we use the fact that groups withisomorphic power graphs also have isomorphic enhanced power graphs, andso have equal GK graphs, by what has already been proved. (cid:3) G and H aregroups for which Com( G ) is isomorphic to EPow( H ). Then the Gruenberg–Kegel graphs of G and H are equal.I do not know whether the analogous result holds for the non-generatinggraph of a non-abelian 2-generated group.The Gruenberg–Kegel graph is an active topic of research; see [39] for asurvey and some recent results. Some of the research concerns the question ofwhether a group is determined (perhaps up to finitely many possibilities) byits GK graph. There are two versions of this: two GK graphs could be equal(as graphs whose vertex set is a finite set of primes) or merely isomorphic asgraphs but with possibly different labels for the vertices. For an interestingexample, the GK graphs of the groups A and Aut(J ) are isomorphic, andboth have vertex sets { , , , } , but are not equal: the labels 2 and 3 areswapped. Let G be a finite group, not trivial and not a cyclic group of prime order. The intersection graph of G is the graph whose vertices are the non-trivial propersubgroups of G , with two vertices H and H adjacent if H ∩ H = { } .There are various other intersection graphs: we can restrict to subgroupsin a particular class, or to maximal subgroups.We will see a connection between some intersection graphs and some ofthe graphs in our hierarchy. For a non-abelian finite group G , there are seven graphs in the hierarchy,and a natural question is: When can two of them be equal? If they are notequal, what can be said about their difference? At the two ends, things are easy:
Proposition 3.1 (a)
Pow( G ) is equal to the null graph if and only if G isthe trivial group. b) NGen( G ) is equal to the complete graph if and only if G is not -generated.(c) NGen( G ) = Com( G ) if and only if G is either abelian and not -generated, or a minimal non-abelian group. Proof
Parts (a) and (b) are clear. So suppose that NGen( G ) = Com( G )and this is not the complete graph. Then G is non-abelian, but if x and y do not generate G then they commute; so every proper subgroup of G isabelian. Thus G is minimal non-abelian. (cid:3) The minimal non-abelian groups were determined by Miller and Moreno [82]in 1903. There are two types: the first consists of groups of prime power or-der; the second are extensions of an elementary abelian p -group by a cyclic q -group, where p and q are primes.Leaving aside the deep commuting graph from the present, the followingwas shown in [1]: Proposition 3.2
Let G be a finite group.(a) The power graph of G is equal to the enhanced power graph if and onlyif G contains no subgroup isomorphic to C p × C q , where p and q aredistinct primes; equivalently, the Gruenberg–Kegel graph of G is a nullgraph.(b) The enhanced power graph of G is equal to the commuting graph ifand only if G contains no subgroup isomorphic to C p × C p , where p isprime; equivalently, the Sylow p -subgroups of G are cyclic or generalizedquaternion groups.(c) The power graph of G is equal to the commuting graph if and only if G contains no subgroup isomorphic to C p × C q , where p and q are primes(equal or distinct). Proof (a) If G contains commuting elements of orders p and q , they areadjacent in EPow( G ) but not in Pow( G ). Conversely, suppose that x and y are adjacent in EPow( G ) but not in Pow( G ). Then x and y are containedin a cyclic group C but neither is a power of each other; C must then haveorder divisible by two distinct primes.(b) If G contains commuting elements of the same prime order p but notin a cyclic subgroup of order p , they are joined in the commuting graph but18ot in the enhanced power graph. Conversely, suppose that x and y areadjacent in Com( G ) but not in EPow( G ). The orders of x and y must havea common factor (otherwise they generate a cyclic group); so some powersof them have prime order p and generate C p × C p .Now a theorem of Burnside (see [59, Theorem 12.5.2]) shows that a p -group containing no subgroup C p × C p is cyclic or generalized quaternion.(c) The third part is immediate from the first two. (cid:3) Using these results it is possible to classify the groups involved.(a) Any group of prime power order has Gruenberg–Kegel graph consistingof a single vertex, so has power graph equal to enhanced power graph.Any other group with this property has disconnected Gruenberg–Kegelgraph, and so satisfies the conclusion of Theorem 2.8. These groups arethe ones with the property that every element has prime power order;they are sometimecs called
EPPO groups . They, and some generali-sations, were studied intensively in the 1960s and 1970s by GrahamHigman and his students in Oxford (see for example [64]). Recently,Cameron and Maslova [39] have given a complete list of EPPO groups.(b) A group with all Sylow subgroups cyclic is metacyclic; indeed, if theprimes dividing its order are p , p , . . . , p r in increasing order, then ithas a normal Hall subgroup corresponding to the last i primes in thislist, for 1 ≤ i ≤ r − Z ∗ -theorem [53], if G has generalized quaternion Sy-low 2-subgroup, and O ( G ) is the largest normal subgroup of odd orderin G , then G/O ( G ) has a unique involution; the quotient ¯ G by the sub-group generated by this involution has dihedral Sylow 2-subgroup, sofalls into the classification by Gorenstein and Walter [55]. Of the groupsin their theorem, we retain only those with cyclic Sylow subgroups forodd primes, that is, ¯ G is isomorphic to PSL(2 , p ) or PGL(2 , p ) or two adihedral 2-group. Conversely, each such group can be lifted to a uniquegroup with a unique involution. The normal subgroup O ( G ) has all itssubgroups cyclic, so is metacyclic, as above.Finally, the deep commuting graph lies between the enhanced powergraph and the commuting graph. In order to investigate equality here, weneed another construction. Recall that the Schur multiplier of G is the largest19ernel Z in a stem extension H of G (with Z ≤ Z ( H ) ∩ H ′ and H/Z ∼ = G ).An extension is said to be commutation-preserving , or CP, if whenever twoelements x, y ∈ G commute, their preimages in H also commute. Now thereis a well-defined largest kernel of a CP stem extension of G ; this is the Bo-gomolov multiplier of G , see [23, 69].The Bogomolov multiplier first arose in connection with the work of Artinand Mumford on obstructions to Noether’s conjecture on the pure transcen-dence of the field of invariants; it is also connected with other topics innumber theory such as the Tate–Shafarevich set, and in group theory suchas the coclass. However, we only need the definition given above. Proposition 3.3
Let G be a finite group.(a) DCom( G ) = EPow( G ) if and only if G has the following property: let H be a Schur cover of G , with H/Z = G . Then for any subgroup A of G , with B the corresponding subgroup of H (so Z B and B/Z = A ),if B is abelian, then A is cyclic.(b) DCom( G ) = Com( G ) if and only if the Bogomolov multiplier of G isequal to the Schur multiplier. I refer to [36] for the proofs.A precise characterisation of the groups attaining either equality is notknown; but examples exist where one bound but not the other is met, orwhere neither bound is met (see [36]): • If G is the symmetric or alternating group of degree at least 8, then E (EPow( G )) ⊂ E (DCom( G )) ⊂ E (Com( G )). • If G is a dihedral group of order 2 n with n ≥
3, then E (EPow( G ) = E (DCom( G ) ⊂ E (Com( G )). • If G is a certain group of order 64 (number 182 in the GAP library),then E (EPow( G )) ⊂ E (DCom( G )) = E (Com( G )).Note that • if the Schur multiplier of G is trivial, then DCom( G ) = Com( G ); • in general, the Bogomolov multiplier is much smaller than the Schurmultiplier; for example, if G is a non-abelian finite simple group, thenits Bogomolov multiplier is trivial [74].20 uestion 4 (a) What can be said about groups G for which DCom( G ) =EPow( G )?(b) What can be said about groups G for which DCom( G ) = Com( G )? For any pair of graphs in the hierarchy, if G is a group such that these twographs are unequal, we could ask about the graph whose edge set is thedifference. We could denote these by using, for example, (Com − Pow)( G )for the graph whose edges are those belonging to the commuting graph butnot the power graph, with similar notation in other cases.At the top, the difference between the complete graph and the non-generating graph is just the generating graph, which has been extensivelystudied. At the next level, the difference between the generating graph andthe commuting graph (the graph (NGen − Com)( G )) has been studied bySaul Freedman; the results will appear in his thesis. The most completeresults are for nilpotent groups, and are reported in [32]. In particular, if G is nilpotent and the non-commuting non-generating graph is not null, thenafter deletion of all isolated vertices it is connected, with diameter 2 or 3.Other differences (apart from the difference between the power graph andthe null graph) have not been studied. Question 5
For each pair of graph types in the hierarchy, what can be saidabout groups for which the difference is connected (after removing isolatedvertices and vertices joined to all others)?In section 9, I give a very weak partial result on the graph (Com − Pow)( G ). We saw the result of [109] that two groups have isomorphic power graphs ifand only if they have isomorphic enhanced power graphs.
Question 6
Are there any other implications of this kind between pairs ofgraphs in the hierarchy?For a simple negative example, the groups C p and C p × C p have isomor-phic commuting graphs but nonisomorphic power graphs, while the group21 p × C p × C p and the non-abelian group of order p and exponent p haveisomorphic power graphs but nonisomorphic commuting graphs.Do there exist groups G and G such that, for example, Pow( G ) isisomorphic to Com( G )? This will be true if Pow( G ) = Com( G ) and G and G have isomorphic commuting graphs, or if Pow( G ) = Com( G ) and G and G have isomorphic power graphs. Question 7
Can Pow( G ) and Com( G ) be isomorphic for groups G and G which both have power graph not equal to commuting graph? Similarquestions for other pairs of graphs in the hierarchy. It is clear that, if Γ and Γ share a vertex set and E (Γ ) ⊆ E (Γ ), then ω (Γ ) ω (Γ ), χ (Γ ) χ (Γ ), α (Γ ) > α (Γ ), and θ (Γ ) > θ (Γ ). So thesefour parameters are monotonic for the graphs in the hierarchy on a givengroup (non-decreasing for ω and χ , non-increasing for α and θ ). Proposition 4.1 (a) ω (EPow( G )) is the order of the largest cyclic sub-group of G ;(b) ω (Com( G )) is the order of the largest abelian subgroup of G ;(c) ω (DCom( G )) is the order of the largest subgroup of G whose inverseimage in any central extension of G is abelian. Proof (a) and (b) follow from Propositions 2.1 and 2.4. For (c), apply (b)to a Schur cover of G . (cid:3) The power graph of a group G is perfect, and so has equal clique numberand chromatic number. These numbers do not exceed the clique number ofthe enhanced power graph, which is the largest order of an element of G ; butthey may be smaller. For example, ω (Pow( C )) = 5, but EPow( C ) is thecomplete graph K .As noted, ω (Pow( G )) ω (EPow( G )). There is an inequality in the otherdirection: Theorem 4.2
There is a function f on the natural numbers such that, forany finite group G , ω (EPow( G )) f ( ω (Pow( G ))) . roof Let m be the largest prime power which is the order of an element of G . The power graph of an m -cycle is complete; so ω (Pow( G )) > m . On theother hand, the largest order of an element of G is not greater than the leastcommon multiple of { , . . . , m } , say f ( m ); and so ω (EPow( G )) f ( m ). (cid:3) No such result holds for the commuting graph. If G is an elementaryabelian group of order 2 n , then ω (EPow( G )) = 2 but ω (Com( G )) = 2 n .The function f in the theorem is exponential. For there are π ( m ) (1 + o (1)) m/ log m primes up to m , and f ( m ) is the product of the largestpower of each prime not exceeding m ; so f ( m ) m (1+ o (1)) m/ log m = e (1+ o (1)) m .But the true value is probably much smaller. Question 8
Find the best possible function f in Theorem 4.2.An upper bound for the chromatic number of the enhanced power graphis given in [1, Theorem 12]: Theorem 4.3
Let G be a finite group, and S the set of orders of elementsof G . Then χ (EPow( G )) ≤ X n ∈ S φ ( n ) , where φ is Euler’s function. Proof
For each n ∈ S , the set of elements of order n in G is the disjointunion of complete graphs of size φ ( n ), and can be coloured with φ ( n ) colours.If we use disjoint sets of colours for different orders, no further clashes willoccur. (cid:3) The bound is met for abelian groups. For if G is abelian, then S is the setof divisors of the exponent of G , and so the sum on the right is the exponent,which is the largest element order in G . Question 9
Find a formula for the clique number of the power graph, orthe chromatic number of the enhanced power graph, of a finite group.Results about the independence number and clique cover number are lesswell developed. Since the power graph is perfect, the Weak Perfect GraphTheorem of Lov´asz [78] asserts that its complement is also perfect, so α (Pow( G )) = θ (Pow( G ))for any finite group G . (Alternatively this follows from Dilworth’s Theorem.)The independence number of the non-generating graph of a finite grouphas been investigated by Lucchini and Mar´oti [79].23 Induced subgraphs
In this section I will consider the question, for each of the graphs in ourhierarchy: For which finite graphs Γ does there exist a finite group G suchthat Γ is isomorphic to an induced subgraph of the group of that type definedon G ? (An induced subgraph of Γ on a subset A of the vertex set consists ofthe vertices of A and all edges of Γ which are contained in A .)To summarise the results: • A finite graph Γ is isomorphic to an induced subgraph of the powergraph of some finite group G if and only if Γ is the comparabilitygraph of a partial order. • For each of the other graphs in the hierarchy, every finite graph isisomorphic to an induced subgraph of that graph defined on some finitegroup.Three related questions are:
Question 10 (a) What is the smallest group for which a given graph isembeddable in the enhanced power graph/deep commuting graph/commutinggraph/non-generating graph?(b) What is the smallest group in which every graph on n vertices can beembedded in one of these graphs?(c) Which graphs occur if we restrict the group to have a particular prop-erty such as nilpotence or simplicity?Here is a very rough lower bound for the smallest N such that every n -vertex graph can be embedded in the enhanced power graph, deep commutinggraph, commuting graph, or non-generating graph of some group of order atmost N . For our rough calculation, we need only consider groups of order N . It is known that there are at most 2 c (log N ) such groups (see [21]); eachhas at most N n subsets of size n . But there are at least 2 n ( n − / /n ! graphson n vertices up to isomorphism. So we require2 c (log N ) · N n ≥ n ( n − / /n ! , which implies that N ≥ n / − ǫ . So the exponential bound we find in somecases is not too far from the truth. 24ote also that every n -vertex graph is embeddable in a Paley graph oforder q , where q is a prime power congruent to 1 (mod 4) and q > n n − (see [22, 24]); so, to find a group whose commuting graph, etc., embeds allgraphs of order n , we only need to embed this Paley graph. Theorem 5.1
Every finite graph is isomorphic to an induced subgraph of thecommuting graph of a finite group. This group can be taken to be nilpotentof class and exponent . Proof
Let F be the two-element field, V a vector space over F , and B abilinear form on V . Define an operation ◦ on V × F by the rule( v , a ) ◦ ( v , a ) = ( v + v , a + a + B ( v , v )) . It is a straightforward exercise to show that this operation makes V × F agroup. This group is nilpotent of class 2 and exponent (dividing) 4, since { } × F is a central subgroup with elementary abelian quotient. Moreover,( v , a ) and ( v , a ) commute if and only if B ( v , v ) = B ( v , v ).Now a bilinear form is uniquely determined by its values on pairs ofvectors taken from a basis for V ; these values can be assigned arbitrarily.So let Γ be a graph with vertex set { , . . . , n } , and let v , . . . , v n be a basis.Assign the values B ( v i , v j ) = 0 if i j ; for i > j , put B ( v i , v j ) = 0 if vertices i and j are adjacent, 1 if not. Then it is clear that the induced subgraph ofthe commuting graph on the set { v , . . . , v n } is isomorphic to Γ. (cid:3) The construction above shows that the smallest group whose commutinggraph contains a given n -vertex graph has order at most 2 n +1 if Γ has n vertices. However, it may be very much smaller; for the complete graph K n ,the answer is clearly n . Theorem 5.2
Every finite graph is isomorphic to an induced subgraph ofthe deep commuting graph of a finite group.
Proof
Let Γ be a finite graph. As we have seen, Γ is isomorphic to aninduced subgraph of the commuting graph of some group. So it is enough to25how that this group can be chosen to have trivial Schur multiplier. Sincethe induced subgraph on a subgroup H of the commuting graph of G is thecommuting graph of H , it suffices to show that every finite group can beembedded in a finite group with trivial Schur multiplier.By Cayley’s Theorem, every finite group of order n can be embeddedin the symmetric group S n . Unfortunately the symmetric group has Schurmultiplier C if n >
8. So we embed S n into the general linear group GL( n, n . Now the Schur multiplier of GL( n,
2) istrivial except for n = 3 or n = 4 [56]. (cid:3) We saw that the power graph of G is the comparability graph of a partial or-der; hence any induced subgraph is also a comparability graph. The converseis also true: Theorem 5.3
A finite graph is isomorphic to an induced subgraph of thepower graph of a finite group if and only if it is the comparability graph of apartial order. The group can be taken to be cyclic of squarefree order.
Proof
One way round follows from our preliminary remarks: the powergraph is the comparability graph of a partial order, and the class of suchgraphs is closed under taking induced subgraphs.So suppose that we have a partial order on X . For each x ∈ X , let[ x ] = { y ∈ x : y x } . A routine check shows that • [ y ] ⊆ [ x ] if and only if y x ; • [ x ] = [ y ] if and only if x = y .So the given partial order is isomorphic to the set of subsets of X of the form[ x ], ordered by inclusion.Now choose distinct prime numbers p x for x ∈ X . Let G be the directproduct of cyclic groups C p x = h a x i of order p x for x ∈ X . Map the subset Y of X to the element g X = ( g x : x ∈ X ) of the direct product, where g x = n a x if x ∈ Y ,1 otherwise.It is readily checked that g X and g Y are adjacent in the power graph if andonly if X and Y are adjacent in the comparability graph of the inclusionorder on X .To conclude, we note that G is a cyclic group of squarefree order. (cid:3) Theorem 5.4
Every finite graph is isomorphic to an induced subgraph ofthe enhanced power graph of some group (which can be taken to be abelian).
Proof
The proof is by induction. For a graph with a single vertex, thereis no problem. So let Γ be a graph with vertex set { , . . . , n } , and supposethat i x i (for i = 1 , . . . , n −
1) is an isomorphism to an induced subgraphof EPow( G ).Choose a prime p not dividing the order of G , and let H = h a, b i be anelementary abelian group of order p . Now in the group G × H , replace x i by x i a if i is not joined to n in Γ, and leave it as is if i is joined to n . Thenmap n to x n = b .Since p ∤ | G | , for any z ∈ G we have h z, a i = h z i × h a i , which is cyclic.So the embedding of { , . . . , n − } is still an isomorphism to an inducedsubgraph. Moreover, h x i , b i is cyclic while h x i a, b i is not, so we have thecorrect edges from b to the other vertices, and the result is proved.The resulting group is the product of n copies of C p × C p for distinctprimes p . (cid:3) The dual enhanced power graph is even easier:
Theorem 5.5
Every finite graph is isomorphic to an induced subgraph of thedual enhanced power graph of some group (which can be taken to be cyclic).
Proof
Let E be the edge set of Γ; choose distinct primes p e for each edge e ∈ E , and let C p e = h a e i be the cyclic group of order p e , and G the directproduct of all these cyclic groups. Now represent a vertex v by the element b v = Q v ∈ e a e . Then h b v i ∩ h b e i = (cid:26) h a e i if e = { v, w } , { } otherwise.So Γ is an induced subgraph of DEP( G ). (cid:3) .5 The generating graph Theorem 5.6
Every finite graph is isomorphic to an induced subgraph ofthe generating graph of a finite group.
Proof
Let Γ be a finite graph. We proceed in a number of steps.
Step 1
Replace Γ by its complement.
Step 2
Every graph can be represented as the intersection graph of a linear hypergraph , a family of sets which intersect in at most one point (whereintersection 1 corresponds to adjacency). The ground set E is the set of edgesof the graph; the vertex v is represented by the set S ( v ) of edges incidentwith v . Then for distinct vertices v, w , S ( v ) ∩ S ( w ) = (cid:26) e, if { v, w } is an edge e , ∅ if v and w are nonadjacent. Step 3
Add some dummy points, each lying in just one of the sets,so that they all have the same cardinality k , with k ≥
3. Now add somedummy points in none of the sets so that the cardinality n of the set Ω ofpoints satisfies the conditions that n > k and n − k is prime. Step 4
Now replace each set by its complement. The complements oftwo subsets of Ω have union Ω if and only if the two sets are disjoint. Thus,each original vertex is now represented by an ( n − k )-set where two such setshave union Ω if and only if the corresponding vertices are adjacent in Γ. Step 5
Replace each set by a cyclic permutation on that set, fixing theremaining points. Each of these cycles has odd prime length, so each is aneven permutation, and so lies in the alternating group A n . Let g v be thepermutation corresponding to the vertex v of Γ. • If v and w are nonadjacent, then the supports of g v and g w have unionstrictly smaller than Ω, so h g v , g w i 6 = A n . • Suppose v and w are adjacent. Then the supports of g v and g w haveunion Ω, so H = h g v , g w i is transitive on Ω. It is primitive: for eachof g v and g w is a cycle of prime length n − k > n/
2, and a block of28mprimitivity either contains the cycle (and so has length greater than n/
2, hence n ) or meets it in one point (and so there are more than n/ n blocks). Hence H is a primitive group of degree n containing a cycle of prime length p with n/ < p < n −
2, By Jordan’stheorem [103, Theorem 13.9], H contains the alternating group A n .Since it is generated by even permutations, H = A n .Thus we have embedded Γ as an induced subgraph in the generatinggraph of A n , as required. (cid:3) We can also ask which graphs can be embedded in the graph whose edge setis the difference of the edge sets of two graphs in the hierarchy.The proof that enhanced power graphs are universal uses abelian groupsfor the embedding. So, by embedding the complement, it shows:
Corollary 5.7
Let Γ be a finite graph. Then there is a group G such that Γ is isomorphic to an induced subgraph of Com − EPow( G ) . However, a much stronger result is true:
Theorem 5.8
Let Γ be a finite complete graph, whose edges are colouredred, green and blue in any manner. Then there is an embedding of Γ into afinite group G so that(a) vertices joined by red edges are adjacent in the enhanced power graph;(b) vertices joined by green edges are adjacent in the commuting graph butnot in the enhanced power graph;(c) vertices joined by blue edges are non-adjacent in the commuting graph. Proof
We begin with two observations. First, the direct product of cyclic(resp. abelian) groups of coprime orders is cyclic (resp. abelian).Second, consider the non-abelian group of order p and exponent p , where p is an odd prime: P = h a, b | a p = b p = 1 , [ a, b ] = a p i . h a i generate a cyclic group; and the group generated by b and x is cyclic if x = 1, abelian but not cyclic if x = a p , and non-abelian if x = a .The proof is by induction on the number n of vertices. The result is clearlytrue if n = 1. So let { v , . . . , v n } be the vertex set of Γ, and suppose that wehave an embedding of { v , . . . , v n − } into a group G satisfying (a)–(c).Choose an odd prime p not dividing | G | , and consider the group P × G ,where P is as above. Modify the embedding of the first n − v i by (1 , v i ) if { v i , v n } is red, by ( a p , v i ) if { v i , v n } is green, and by( a, v i ) if { v i , v n } is blue. It is easily checked that we still have an embeddingof { v , . . . , v n − } satisfying (a)–(c). Moreover, if we now embed v n as ( b, (cid:3) Clearly there are plenty of problems along similar lines to investigate here.
There are a number of graph products. Here I will be chiefly concerned withthe strong product, defined as follows.Let Γ and ∆ be graphs with vertex sets V and W respectively. The strongproduct Γ ⊠ ∆ has vertex set the Cartesian product V × W ; vertices ( v , w )and ( v , w ) are joined whenever v is equal or adjacent to v and w is equalor adjacent to w , but not equality in both places. (All of the graphs in thehierarchy naturally have loops at each vertex, which we have discarded; thestrong product is the natural categorical product in the category of graphswith a loop at each vertex.)I note that the strong product, along with the Cartesian and categoricalproducts, is denoted by a symbol representing the corresponding product oftwo edges: the Cartesian product is Γ (cid:3) ∆, while the categorical product isΓ × ∆.The question of the perfectness of strong products of graphs has beenstudied by Ravindra and Parthasarathy [89].The only group product that concerns us here is the direct product. Proposition 6.1
Let G and H be finite groups.(a) Com( G × H ) = Com( G ) ⊠ Com( H ) . b) If G and H have coprime orders, then EPow( G × H ) = EPow( G ) boxtimes EPow( H ) .(c) If G/G ′ and H/H ′ have coprime orders, and in particular if G and H are perfect groups, then DCom( G × H ) = DCom( G ) ⊠ DCom( H ) . Proof (a) Distinct elements ( g , h ) and ( g , h ) in G × H commute if andonly if g and g are equal or commute, and h , h are equal or commute.(b) Suppose that | G | and H are coprime. If h g , g i and h h , h i arecyclic, then (as their orders are coprime) their direct product is also cyclicand contains ( g , h ) and ( g , h ). Conversely, again using coprimeness, if h ( g , h ) , ( g , h ) i is cyclic, then it contains ( g , g , , h ) and (1 , h ).(c) A formula of Schur gives the Schur multiplier of G and H to be M ( G ) × M ( H ) × ( G ⊗ H ), where M ( G ) is the Schur multiplier of G . If | G/G ′ | and | H/H ′ | are coprime, the third term is absent. It follows that aSchur cover of G × H is the direct product of Schur covers of G and H . Theresult follows. (cid:3) Thus, questions about the commuting graph or enhanced power graphof a nilpotent group can be reduced to questions about the correspondinggraphs for their Sylow subgroups.The corresponding result fails for the power graph and the non-generatinggraph. The power graphs of C and C are complete but the power graph of C × C are not. For the non-generating graph, we note that for any non-abelian finite simple group G , there is an integer m such that G n fails to be2-generated if n > m . Two vertices in a graph are called twins if they have the same neighbours(possibly excluding one another). Equivalently, v and w are twins if thetransposition ( v, w ) (fixing the other vertices) is an automorphism of Γ. If G is a non-trivial group, then any of the graphs in our hierarchy based on G will contain many pairs of twins. Thus, twins will play an important partwhen we come to look at automorphism groups.If a graph has twins, then we can make a new graph by merging thetwins to a single vertex. The process can be continued until no pairs of twins31emain. If the resulting graph has just a single vertex, the original graph iscalled a cograph.Cographs also play an important part in the story, and make anotherlink with the Gruenberg–Kegel graph. So we make a detour to look at twinreduction and cographs. A graph Γ is a cograph if either of the following equivalent conditions holdsfor it: • Γ does not contain the four-vertex path P as an induced subgraph; • Γ can be constructed from the 1-vertex graph by the operations ofcomplement and disjoint union.In particular, a cograph is connected if and only if its complement is discon-nected. This leads to a tree representation of cographs and to very efficientalgorithms for determining their properties.I mention here a curious fact which may (or may not) have a connectionwith the following material. The P -structure of a graph Γ is the hypergraphwhose vertex set is the same as that of Γ, the hyperedges being the 4-elementsets which induce a copy of P in Γ. We have the following easy observations: • a graph and its complement have the same P -structure, since the graph P is self-complementary; • a graph is a cograph if and only if its P -structure is the null hyper-graph.Bruce Reed [90] proved the semi-strong perfect graph theorem , which hadbeen conjectured by Vasek Chv´atal, asserting that if two graphs have iso-morphic P -structures and one is perfect, then so is the other.Cographs have been rediscovered a number of times, and as a resultappear in the literature with very different names, such as “complement-reducible graphs”, “hereditary Dacey graphs” and “N-free graphs”. See[94, 99, 70] for information about cographs.In a graph Γ, we can define two kinds of “twin relations” on vertices.The open neighbourhood Γ( v ) of v in Γ is the set of vertices in Γ joined to v ; the closed neighbourhood is Γ( v ) ∪ { v } . Two vertices v, w are open twins if they have the same open neighbourhoods; they are closed twins if they32ave the same closed neighbourhoods. Both of these relations are obviouslyequivalence relations. Two vertices are open twins in Γ if and only if they areclosed twins in the complement of Γ. Note that open twins are not joined,while closed twins are joined.For either of these relations, we can define a reduced graph by collapsingeach equivalence class to a single vertex. The original graph can be recon-structed uniquely from the reduced graph and the partition into equivalenceclasses. We call the two reductions open twin reduction and closed twin re-duction respectively. In the graph obtained by open twin reduction, the opentwin relation is the relation of equality, and similarly for the closed relation.So, given any graph, we can apply the two reductions alternately until theresulting graph has both twin relations equal to the relation of identity. Wecall such a graph the cokernel of the original graph, for reasons which aremade clear by the following result. Note that, in the most general form oftwin reduction, each move simply identifies one pair of twins, and these canbe open or closed twins arbitrarily. Theorem 7.1
Given a graph Γ , the result of performing a sequence of twinreductions until the graph is twin-free is unique up to isomorphism, indepen-dent of the chosen sequence. Proof
Open and closed twin classes of sizes greater than 1 are disjoint.For suppose that x and y are open twins and y and z are closed twins.Then xy is a non-edge while yz is an edge. Since y and z are twins, x is not joined to z ; but since y and x are twins, z is joined to x . Theseconclusions are contradictory. (Alternatively, since y and z are twins, ( y, z )is an automorphism, and so x and z are open twins; and similarly using theautomorphism ( x, y ), x and z are closed twins.)We are going to prove the theorem by induction on the number of vertices.There is nothing to do for graphs with a single vertex, so let Γ have n vertices,with n >
1, and assume that the result is true for any graph with fewer than n vertices. Take two twin reduction sequences on Γ. Suppose that the firstbegins by identifying x and y , and the second by identifying u and v .If { x, y } = { u, v } , then the two sequences result in the same graph, andinduction finishes the job. 33f |{ x, y } ∩ { u, v }| = 1, then our initial remark shows that the two pairsof twins have the same type, so the graphs obtained after one step are iso-morphic, and again induction finishes the job.Suppose that { x, y } ∩ { u, v } = ∅ . Then the two reductions commute. Let∆ be the graph obtained by applying the two reductions. Then ∆ occursafter two steps in reduction sequences for Γ beginning by identifying x and y , or by identifying u and v . By induction the end result of either givensequence is the same as the result of reducing ∆ (up to isomorphism).The theorem is proved. (cid:3) The next result gives the connection between cographs and twin reduc-tion.
Proposition 7.2
A graph Γ is a cograph if and only if the cokernel of Γ isthe graph with a single vertex. Proof
For the necessity, we show by induction that a cograph with morethan one vertex contains twins. Let Γ be a cograph with more than onevertex, and suppose that any smaller cograph with more than one vertexcontains twins. If Γ is disconnected, then if it has a component with morethan one vertex, then this component contains twins; otherwise Γ is a nullgraph and all pairs of vertices are open twins. If Γ is connected, then itscomplement is disconnected, and we argue in the complement instead.Now the result of twin reduction is an induced subgraph of Γ, and so alsoa cograph; so so the reduction continues until only one vertex remains.Conversely, suppose that Γ is not a cograph. Then Γ contains a 4-vertexpath, say ( w, x, y, z ). Then any pair of these vertices are not twins, and soare not identified in any twin reduction; so the result of the reduction stillcontains a 4-vertex path. So no sequence of reductions can terminate in asingle vertex. (cid:3)
This gives another test for a cograph: apply twin reductions until theprocess terminates, and see whether just one vertex remains.
The relevance of this to our problem is:34 roposition 7.3
Let Γ be the power graph, enhanced power graph, deepcommuting graph, commuting graph, or non-generating graph of a non-trivialgroup G . Then the twin relation on Γ is not the relation of equality. Proof
Suppose that G contains an element g of order greater than 2. Let h be an element such that g = h but h g i = h h i . Then any element joined toone of g and h in one of the graphs listed is also joined to the other. Thearguments are all easy; let us look at the least trivial, the deep commutinggraph. Let H be a Schur cover of G with kernel Z , and x and y elements of H covering g and h respectively. Then h Z, x i is abelian and contains y , so x and y commute.The groups not covered by this are elementary abelian 2-groups. Inthese cases, everything is clear: the power graph, enhanced power graphand deep commuting graph are stars; the commuting graph is complete; thenon-generating graph is complete if the group has order greater than 4. Allthese graphs are cographs. (cid:3) So, for any of our classes of graphs, say X, the question “What is thecokernel of X( G )?” is a generalisation of “Is X( G ) a cograph?”Finally for this section, I note that the class of cographs is not preservedby strong product. Figure 2 shows P as an induced subgraph of P ⊠ P . r r rr r rr r r (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)❅❅❅❅❅❅❅❅❅❅❅❅ ❡ ❡ ❡❡ ❅❅❅ Figure 2: Cographs are not closed under ⊠ In this section we consider various classes of graphs defined by forbiddeninduced subgraphs, and ask: for which finite groups G can one of the graphsin the hierarchy defined on G belong to this class? Not much is known.We saw that a graph is perfect if and only if it has no induced subgraphisomorphic to an odd cycle of length greater than 3 or the complement of35ne. In particular, a cograph (which contains no induced 4-vertex path) isperfect.We have seen that power graphs are comparability graphs of partial or-ders, and noted that these graphs are perfect. For our other types of graph,no proper subclass defined by forbidden induced subgraphs contains themall; so we have to ask a different question: which groups have the propertythat one of these graphs belongs to a graph class defined in this way? We begin with the question: When is the power graph of the group G a co-graph? This question was considered in the paper [38]; here the Gruenberg–Kegel graph makes another appearance. The question is answered completelyfor nilpotent groups, but a necessary and a sufficient condition are known forgeneral groups in terms of the GK graph; these conditions do not coincide,and we will see that no condition just in terms of the GK graph can be bothnecessary and sufficient. Theorem 8.1
Let G be a finite nilpotent group. Then Pow( G ) is a cographif and only if either(a) G has prime power order; or(b) G is cyclic of order pq , where p and q are distinct primes. Theorem 8.2 (a) Let G be a finite group whose Gruenberg–Kegel graph isa null graph. Then the power graph of G is a cograph.(b) Let G be a finite group whose power graph is a cograph. Then, withpossibly one exception, a connected component of the Gruenberg–Kegelgraph has at most two vertices, the exception being the component con-taining the prime . If { p, q } is a connected component of the GKgraph, with p and q odd primes, then p and q divide | G | to the firstpower only. For the final claim, note that the groups PSL(2 ,
11) and M have thesame GK-graph (an edge { , } and isolated vertices { } and { } ), but thepower graph of the first is a cograph, that of the second is not. Question 11
Classify the groups whose power graph is a cograph.36he remainder of this subsection is based on the paper [38].We saw that the power graph of a finite group is a comparability graphof a partial order, and so in particular is a perfect graph. Various interestingsubclasses of the perfect graphs are defined by forbidding certain inducedsubgraphs. Cographs form an example: they forbid the 4-vertex path P .Here are some other graph classes. • A graph is chordal if it contains no induced cycles of length greaterthan 3 (that is, every cycle of length greater than 3 has a chord). • A graph is split if the vertex set is the disjoint union of two subsets, oneinducing a complete graph and the other a null graph (with possiblysome edges between them). A graph is split if it contains no inducedsubgraph isomorphic to C , C or 2 K . • A graph is threshold if it can be constructed from the 1-vertex graph byadding vertices joined either to all or to no existing vertices. A graphis threshold if and only if it contains no induced subgraph isomorphicto P , C , or 2 K . Theorem 8.3
For a finite nilpotent group G , the power graph of G is chordalif and only if one of the following conditions holds:(a) G has prime power order;(b) G is the direct product of a cyclic group of p -power order and a groupof exponent q , where p and q are distinct primes. Question 12
Which non-nilpotent groups have the property that the powergraph is chordal?
Theorem 8.4
The following conditions for a finite group are equivalent:(a)
Pow( G ) is a threshold graph;(b) Pow( G ) is a split graph;(c) Pow( G ) contains no induced subgraph isomorphic to K ;(d) G is cyclic of prime power order, or an elementary abelian or dihedral -group, or cyclic or order p , or dihedral of order p n or p , where p is an odd prime. Note that this theorem does not assume that G is nilpotent.37 .2 Other graphs For our other classes of graphs, the problem of deciding for which groupsthe graph in question forbids a certain induced subgraph has not been muchworked on, and a number of interesting questions are open. One differenceis that, as we have seen, every finite graph occurs as an induced subgraph ofeach type of graph on some finite group.In line with the preceding subsection we could ask a multipart question:
Question 13
For which finite groups is the enhanced power graph/deepcommuting graph/commuting graph/nongenerating graph a perfect graph,or a cograph, or a chordal graph, or a split graph, or a threshold graph?We can give a partial answer for groups of prime power order.
Theorem 8.5
Let G be a group of prime power order.(a) The power graph of G is equal to the enhanced power graph, and con-tains no induced P or C .(b) If G is -generated of order p n , then the non-generating graph of G consists of p + 1 complete subgraphs of order p n − , any two intersectingin the same subset of size p n − . Proof (a) In a group of prime power order, if two elements generate a cyclicsubgroup, then one is a power of the other; this shows that the power graphand enhanced power graph coincide. Suppose that ( x, y, z ) is an inducedpath of length 2. If x, z ∈ h y i , or if y ∈ h x i and z ∈ h y i , or if x ∈ h y i , y ∈ h z i , then x and z are joined, a contradiction. So y ∈ h x i ∩ h z i . Now if w is joined to x but not to y , then ( w, x, y ) is an induced path of length 2, so x ∈ h y i by the same argument. But then y ∈ h z i , so x ∈ h z i , a contradiction.(b) Let Φ( G ) be the Frattini subgroup of G . By the Burnside BasisTheorem, Φ( G ) consists of vertices lying in no generating pair and has index p in G ; moreover, the generating pairs are all pairs of elements lying indistinct non-trivial cosets of Φ( G ). (cid:3) No such result holds for the commuting graph; Theorem 5.1 shows thatthe commuting graphs of 2-groups form a universal class. Computationshows that the smallest group whose commuting graph is not a cographis the symmetric group S : the elements (1 , , , , , , , , , ,
4) induce a 4-vertex path. In fact, seven groups of order 32 havecommuting graphs which are not cographs.Question 13 has been considered for the commuting graph by Britnelland Gill [28], who obtained a partial description of groups for which thecommuting graph is a perfect graph. Assuming that G has a component (asubnormal quasisimple subgroup), they determine all possible components ofsuch groups. Question 14
What about other graph classes, for example planar graphs?For planarity, this may not be too hard for graphs in the hierarchy. Sincethe complete graph on 5 vertices is not planar, it follows that G has noelements of order greater than 4. (For all cases except the power graph, acyclic subgroup induces a complete graph, so the claim is clear. In the powergraph, the power graph of a cyclic group of prime power order is complete,and the power graph of C contains a K .) So G is solvable.Indeed, bounding the genus of a graph bounds its clique number, and so(for graphs in the hierarchy) bounds the orders of elements.See [8] for the commuting graph and its complement. Proposition 8.6
Let G = PSL(2 , q ) , with q a prime power and q ≥ .(a) If q is even, then EPow( G ) , DCom( G ) and Com( G ) are cographs; Pow( G ) is a cograph if and only if q − and q + 1 are either primepowers or products of two distinct primes.(b) If q is odd, then EPow( G ) and DCom( G ) are cographs; Pow( G ) is acograph if and only if ( q − / and ( q + 1) / are either prime powersor products of two distinct primes. Proof
We consider the graphs obtained by removing the identity. We notethat a graph Γ is a cograph if and only if the graph obtained by adding avertex joined to all others is a cograph.We begin with the case when q is a power of 2, noting that in this casePSL(2 , q ) has trivial Schur multiplier except when q = 4. The elements ofthe group have order 2 or prime divisors of q − q + 1. The centralisers ofinvolutions are elementary abelian of order q , while the centralisers of other39lements are cyclic of order q − q + 1. In other words, centralisers areabelian and intersect only in the identity. So the commuting graph (which isthe deep commuting graph if q = 4) is a disjoint union of complete graphs;the enhanced power graph is a disjoint union of complete graphs and isolatedvertices (corresponding to the involutions); and the power graph is a disjointunion of the power graphs of cyclic groups of orders q ± C m is a cograph ifand only if m is either a prime power or the product of two primes [38,Theorem 3.2], the result follows. The result for the deep commuting graphfor PSL(2 ,
4) can be proved by computation, or by identifying this groupwith PSL(2 ,
5) (which is dealt with in the next paragraph).Now consider the case when q is a power of an odd prime p . The cen-tralisers of elements of order p are elementary abelian of order q ; centralisersof other elements are cyclic of orders ( q ± / q ±
1, whichever is divisible by 4.So, although centralisers may not be disjoint, the maximal cyclic subgroupsare, so the statements about the enhanced power graph and the commutinggraph are true.If q is odd and q = 9, the Schur multiplier of PSL(2 , q ) is cyclic of order2, and so a Schur cover of this group is SL(2 , q ). The unique involution inthis group is − I , the kernel of the extension; so involutions in PSL(2 , q ) liftto elements of order 4, which lie in unique maximal abelian subgroups.The group PSL(2 ,
9) has Schur multiplier of order 6. The deep commutinggraph of PSL(2 ,
9) was analysed using
GAP , with generators for the Schurcover from the on-line Atlas of Finite Group Representations [105]. (cid:3)
Question 15
Are there infinitely many prime powers q for which the powergraph of PSL(2 , q ) is a cograph?Here is a preliminary analysis of this question. Case q even Then q = 2 d , say, and each of q + 1 and q − q = 8, we conclude that each of q + 1and q − q + 1 and q − − ·
89 while 2 + 1 = 3 · d up to 200 for which q = 2 d satisfies the condition are 1, 2, 3, 4, 5, 7, 11,13, 17, 19, 23, 31, 61, 101, 127, 167, 199. Case q odd If q is congruent to ± q + 1) / q − / q = 9, or q is a Fermat or Mersenne prime. So, apart from this case, q is congruentto ± q is an odd power of 3, or one of ( q + 1) / q − / q = 11 or q = 13). The odd prime powers up to 500 satisfyingthe condition are 3, 5, 7, 9, 11, 13, 17, 19, 27, 29, 31, 43, 53, 67, 163, 173,243, 257, 283, 317.Table 1 gives the numbers of vertices in cokernels for small finite simplegroups. Note that a graph is a cograph if and only if its cokernel has onevertex. The last column of the table gives the number of cyclic subgroups of G , that is, equivalence classes under the relation ≡ where x ≡ y if h x i = h y i .Equivalent vertices are closed twins in all our graphs, so this number is anupper bound for the number of vertices in each cokernel. (I have replacedPSL and PSU by L and U in the table to save space.)The table suggests various conjectures, some of which can be proved. Forexample: Theorem 8.7
Let G be a non-abelian finite simple group. Then NGen( G ) is not a cograph. Proof
We consider the reduced graph obtained by deleting the identityvertex. The reduced non-generating graph of a simple group is connected,and has diameter at most 5 (this follows from the results of Ma, Herzog etal. , Shen and Freedman discussed in Section 12 below). Also, the generatinggraph is connected (see [27]), and indeed has diameter 2 (see [29]). But thecomplement of a connected cograph is disconnected. (cid:3)
Question 16
Find, or estimate, the number of vertices in the cokernel of thenon-generating graph of a finite simple group. In particular, classify simplegroups for which the number of vertices in the cokernel of the non-generatinggraph is equal to the number of cyclic subgroups.41 | G | Pow( G ) EPow( G ) DCom( G ) Com( G ) NGen( G ) Cyc A
60 1 1 1 1 32 32 L (7) 168 1 1 1 44 79 79 A
360 1 1 1 92 167 167 L (8) 504 1 1 1 1 128 156 L (11) 660 1 1 1 112 244 244 L (13) 1092 1 1 1 184 366 366 L (17) 2448 1 1 1 308 750 750 A L (19) 3420 1 1 1 344 914 914 L (16) 4080 1 1 1 1 784 784 L (3) 5616 756 756 808 808 1562 1796 U (3) 6048 786 534 499 499 1346 1850 L (23) 6072 1267 1 1 508 1313 1566 L (25) 7800 1627 1 1 652 1757 2082 M p -group is a cograph (it consists of astar K ,p +1 with the central vertex blown up to a clique of size p n − and theremaining vertices to cliques of size p n − ( p − p n ). This graph is a cograph: its complement is complete multipartite withsome isolated vertices. In this section, we examine the question of connectedness of these graphs.All those in the hierarchy (except the null graph) are connected, since theidentity is joined to all other vertices. The question is non-trivial, however, ifwe remove vertices joined to all others. The first job is to characterise thesevertices. 42 .1 Centres
In the commuting graph of G , the set of vertices joined to all others is simplythe centre Z ( G ) of G . So we adapt the terminology by defining analogues ofthe centre for other graphs in the list. So, if X denotes Pow, EPow, DCom,Com or NGen, we define the X-centre of G , denoted Z X ( G ), to be the set ofvertices joined to all others in X( G ). It turns out that (aside from the non-generating graph), in almost all cases, Z X ( G ) is a normal subgroup of G ; theonly exception is for the power graph of a cyclic group of non-prime-powerorder. We note that Z EPow ( G ) has been studied by Patrick and Wepsic [86],who called it the cyclicizer of G : it is the set { x ∈ G : ( ∀ y ∈ G ) h x, y i is cyclic } . The main results on the cyclicizer are also given in [5].
Theorem 9.1 (a) Z Pow ( G ) is equal to G if G is cyclic of prime powerorder; or the set consisting of the identity and the generators if G iscyclic of non-prime-power order; or Z ( G ) if G is a generalized quater-nion group; or { } otherwise.(b) Z EPow ( G ) is the product of the Sylow p -subgroups of Z ( G ) for p ∈ π ,where π is the set of primes p for which the Sylow p -subgroup of G iscyclic or generalized quaternion; in particular, Z EPow ( G ) is cyclic.(c) Z DCom ( G ) is the projection into G of Z ( H ) , where H is a Schur coverof G .(d) Z Com( G ) = Z ( G ) . Proof (a) This is [31, Proposition 4].(b) Suppose that x is joined to all other vertices in EPow( G ). Then h x, y i is cyclic for all y ∈ G ; so certainly x ∈ Z ( G ).If three elements of a group have the property that any two of themgenerate a cyclic group, then all three generate a cyclic group: see [1, Lemma32]. So Z EPow ( G ) is a subgroup, since if x, y ∈ Z EPow ( G ) then, for all w ∈ G , h x, w i and h y, w i are cyclic, and so if h x, y i = h z i then h z, w i is cyclic for all w ∈ G , so that z ∈ Z EPow ( G ).Now let x be an element of prime order p in Z EPow ( G ). If G containsa subgroup C p × C p then there is an element of order p not in h x i , so notadjacent to x , a contradiction. So the Sylow p subgroup of G is cyclic or43eneralised quaternion, by Burnside’s theorem. But now x lies in every Sylow p -subgroup of G , so is joined to every element of p -power order, and henceto every element of G .(c) and (d) Part (d) is clear, and (c) follows since DCom( G ) is a projectionof the commuting graph of a Schur cover of G . (cid:3) By contrast, Z NGen ( G ) is not necessarily a subgroup of the 2-generatedgroup G . If G is non-abelian, then Z NGen ( G ) must contain Z ( G ), since thenon-generating graph contains the commuting graph. Also, it contains theFrattini subgroup Φ( G ) of G , since 2-element generating sets are minimaland so their elements do not lie in the Frattini subgroup (which consists ofthe elements which can be dropped from any generating set).If the order of G is a prime power, then the Burnside basis theorem showsthat Z NGen ( G ) = Φ( G ), since a set of elements generates G if and only if itsprojection onto G/ Φ( G ) generates this quotient.Also, by the result of [27] (see Theorem 2.5), if all proper quotients of G are cyclic, then Z NGen ( G ) = { } .In general, however, Z NGen ( G ) may be a subgroup different from both Z ( G ) and Φ( G ). For example, let G be the symmetric group S . Then both Z ( G ) and Φ( G ) are trivial, but Z NGen ( G ) is the Klein group V (the minimalnormal subgroup of G ).Moreover, it may not be a subgroup at all. For example, if G = C × C ,then Z NGen ( G ) consists of the elements not of order 6, since both elementsin any generating pair must have order 6. Question 17
Characterise the 2-generated groups in which Z NGen ( G ) is asubgroup of G . Each of our types of graph is connected, since the corresponding “centre” isnon-empty and its vertices are joined to all others. So the question becomesinteresting if we ask whether the induced subgraph on the elements outsidethis centre is connected.The situation for the commuting graph is well-understood, thanks to theresults of [52, 83]. But first I mention another link with the Gruenberg–Kegelgraph. This has been known for some time, but the first mention I know inthe literature is [83, Section 3]. 44 heorem 9.2
Let G be a group with trivial centre. Then the induced sub-graph of the commuting graph on G \ { } is connected if and only if theGruenberg–Kegel graph is connected. Proof
Suppose first that Z ( G ) = 1 and the commuting graph is connected.Let p and q be primes dividing | G | . Choose elements g and h of orders p and q respectively, and suppose their distance in the commuting graph is d . Weshow by induction on d that there is a path from p to q in the GK graph.If d = 1, then g and h commute, so gh has order pq , and p is joined to q .So assume the result for distances less than d , and let g = g , . . . , g d = h bea path from g to h .Let i be mimimal such that p does not divide the order of g i (so i > g i − , say g ai − , has order p , while a power g bi of g i hasprime order r = p .The distance from g bi to g d is at most d − i < d , so there is a path from r to q in the GK graph. But g ai − and g bi commute, so p is joined to r .For the converse, assume that the GK graph is connected.Note first that for every non-identity element g , some power of g hasprime order, so it suffices to show that all elements of prime order lie inthe same connected component of the commuting graph. Also, since a non-trivial p -group has non-trivial centre, the non-identity elements of any Sylowsubgroup lie in a single connected component.Let C be a connected component. Connectedness of the GK graph showsthat C contains a Sylow p -subgroup for every prime p dividing | G | . Also,every element of C , acting by conjugation, fixes C . It follows that the nor-maliser of C is G , and hence that C contains every Sylow subgroup of G ,and thus contains all elements of prime order, as required. (cid:3) After a lot of preliminary work, much of it on specific groups, summarisedin the introduction to [52], Iranmanesh and Jafarzadeh [67] conjectured thatthere is an absolute upper bound on the diameter of any connected compo-nent of the induced subgraph on G \ Z ( G ) of the commuting graph of G . Thisconjecture was refuted by Giudici and Parker [52]. However, it was provedfor groups with trivial centre by Morgan and Parker [83]. In these results, Iuse the term “reduced commuting graph” to mean the induced subgraph ofthe commuting graph of G on G \ Z ( G ), in which no vertex is joined to allothers. 45 heorem 9.3 There is no upper bound for the diameter of the reduced com-muting graph of a finite group; for any given d there is a -group whosereduced commuting graph is connected with diameter greater than d . On the other hand:
Theorem 9.4
Suppose that the finite group G has trivial centre. Then everyconnected component of its reduced commuting graph has diameter at most . For the power graph and enhanced power graph, we note that, if the group G is not cyclic or generalized quaternion, then the corresponding “centre” isjust the identity. So the natural question is: if G is not cyclic or generalizedquaternion, is the induced subgraph of the power graph on non-identity el-ements connected? This question has been considered in several papers, forexample [35, 109].The next result shows that we have only one rather than two problemsto consider. Proposition 9.5
Let G be a group with Z ( G ) = { } . Then the reducedpower graph of G is connected if and only if the reduced enhanced powergraph of G is connected. Proof If g and h are joined in the power graph, they are joined in theenhanced power graph; if they are joined in the enhanced power graph, thenthey lie at distance at most 2 in the power graph; both g and h are powersof the intermediate vertex, which is thus not the identity. (cid:3) The argument shows that, if these graphs are connected, the diameter ofthe power graph is at least as great as, and at most twice, the diameter ofthe enhanced power graph. Can these bounds be improved?I have already quoted the result of Burness et al. on the generating graph.For the non-generating graph, the results of Freedman and others on thedifference between the non-generating graph and the commuting graph (thatis, the graph (NGen − Com)( G )) have been mentioned also.As promised, here is a weak result on the graph (Com − Pow)( G ). Theorem 9.6
Suppose that the finite group G satisfies the following condi-tions:(a) The Gruenberg–Kegel graph of G is connected. b) If P is any Sylow subgroup of G , then Z ( P ) is non-cyclic.Then the induced subgraph of (Com − Pow)( G ) on G \ { } either has anisolated vertex or is connected. Proof
Let Γ( G ) denote the induced subgraph of (Com − Pow)( G ) on G \{ } . Note that, if H is a subgroup of G , then the induced subgraph of Γ( G )on H \ { } is Γ( H ).First we show that, if P is a p -group, then Γ( P ) is connected. Let Q Z ( P ) with Q ∼ = C p × C p . Then the induced subgraph on Q \ { } is completemultipartite with p + 1 blocks of size p −
1, corresponding to the cyclicsubgroups of Q . So it suffices to show that any element z of P \ { } has aneighbour in Q \ { } . We see that z commutes with Q since Q Z ( P ); and h z i ∩ Q is cyclic so there is some element of Q not in this set.Now let C be a connected component of Γ( G ) containing an element z of prime order p . Since Γ( G ) is invariant under Aut( G ), in particular it isnormalized by all its elements, so h C i N G ( C ). In particular, C contains aSylow p -subgroup of G (one containing the given element of order p in C ).If C contains an element of prime order r , and { r, s } is an edge of theGK graph, then G contains an element g of order rs , then without loss ofgenerality g s ∈ C , and g s is joined to g r in Γ( G ), so also g r ∈ C . Nowconnectedness of the GK graph shows that C contains a Sylow q -subgroup of G for every prime divisor of | G | . Hence | N G ( C ) | is divisible by every primepower divisor of | G | , whence N G ( C ) = G .Finally, let g be any non-identity element of G . Choose a maximal cyclicsubgroup K containing g . If C G ( K ) = K , then the generator of K commutesonly with its powers, and is isolated in Γ( G ). If not, then there is an elementof prime order in C G ( K ) \ K . (If h ∈ C G ( K ) \ K , then h g, h i is abelian butnot cyclic, so contains a subgroup h g i × C m for some m ; choose an elementof prime order in the second factor.) This element is joined to g in thecommuting graph but not in the power graph; so g ∈ C . We conclude that C = G \ { } , and the proof is done. (cid:3) Remarks
The theorem is probably not best possible. Let us consider thehypotheses.We saw in Theorem 9.2 that, for groups with trivial centre (our maininterest here), connectedness of the GK graph is equivalent to connectednessof the commuting graph, and so is clearly necessary for connectedness of(Com − Pow)( G ). 47he second condition (which is necessary for the above proof) is probablymuch too strong. Perhaps it can be weakened to say that the Sylow subgroupsof G are not cyclic or generalized quaternion groups (or, subgroups of G arenot cyclic or generalized quaternion groups (or, equivalently, that G has nosubgroup C p × C p for prime p ). Perhaps it is only necessary to assume thisfor one prime. More work needed.
10 Automorphisms
Each type of graph in the hierarchy on a group G is preserved by the auto-morphism group of G . But in almost all cases, the automorphism group ofthe graph is much larger. This question has been considered in [12].We saw in Proposition 7.3 that any of our hierarchy of graphs has non-trivial twin relation. So the first thing we need to do is to take a look atautomorphisms of such graphs.Let Γ be a graph. It is clear that its automorphism group Aut(Γ) pre-serves twin relations on Γ, and that vertices in a twin equivalence class canbe permuted arbitrarily. It follows by induction that the group induces anautomorphism group on the cokernel Γ ∗ of Γ, say Aut − (Γ ∗ ). We say that thetwin reduction on Γ is faithful if Aut − (Γ ∗ ) = Aut(Γ ∗ ).Trivially, if Γ is a cograph (so that its cokernel is the 1-vertex graph), thetwin reduction is faithful; we ignore this case.The reduction process is not always faithful. For a simple example, con-sider Figure 3. r r r rr ✟✟✟❍❍❍ r r r r Figure 3: Non-faithful twin reductionIn the left-hand graph, the two leaves on the right are twins, and twinreduction gives the right-hand graph as the cokernel. But the cokernel hasan automorphism (reflection in the vertical axis of symmetry) not inducedfrom an automorphism of the original.
Question 18
Given a finite group G and one of our types of graph (say X),48a) When is twin reduction on X( G ) faithful?(b) What is the automorphism group of the cokernel of X( G )?Very little seems to be known about this question. I first discuss cographs,then give a couple of examples. Proposition 10.1
Let Γ be a cograph. Then the automorphism group of Γ can be built from the trivial group by the operations of direct product andwreath product with a symmetric group. Proof
Recall that a cograph can be built from the 1-vertex graph by theoperations of complement and disjoint union. Now complementation doesnot change the automorphism group. If Γ is the disjoint union of m copiesof ∆ , . . . , m r copies of ∆ r , thenAut(Γ) = (Aut(∆ ) ≀ S m ) × · · · × (Aut(∆ r ) ≀ S m r ) . Assuming inductively that each of Aut(∆ ), . . . , Aut(∆ r ) can be built bydirect producs and wreath products with symmetric groups; then the sameis true for Aut(Γ). (cid:3) Example
Let G be the alternating group A , and consider the power graphof G . The identity is joined to all other vertices; after removing it we have sixcliques of size 4 (corresponding to cyclic subgroups of order 5), ten of size 2(corresponding to cyclic subgroups of order 3), and fifteen isolated vertices(corresponding to elements of order 2). This graph is easily seen to be acograph, so its cokernel has a single vertex. In fact, closed twin reductioncontracts the cliques of sizes 2 and 4 to single vertices, giving a star on 32vertices; then open twin reduction produces a single edge, and closed twinreduction reduces this to a single vertex. Example
Let G be the Mathieu group M . The power graph of G has7920 vertices. On removing the identity, we are left with a graph consistingof •
144 complete graphs of size 10, corresponding to elements of order 11; •
396 complete graphs of size 4, corresponding to elements of order 5; • a single connected component ∆ on the remaining 4895 vertices.49wo steps of twin reduction remove all the components which are complete.If we take ∆, and first factor out the relation “same closed neighbourhood”,and then factor out from the result the relation “same open neighbourhood”,we obtain a connected graph on 1210 vertices whose automorphism group is M . (This is shown by a GAP computation.) This group is induced by theautomorphism group of the original power graph; so the reduction is faithful.
Exercise
Why is the number 1210 given above two less than the numberof vertices of the cokernel of the power graph of M given in Table 1? Question 19
For which non-abelian finite simple group G is it the casethat the twin reduction on the power graph/enhanced power graph/deepcommuting graph/commuting graph/generating graph of G is faithful? Example
A curious example showing that this is not true for all suchgroups is described in [37]. Let G be the simple group PSL(2 , G is the group PΓL(2 , G ;but the cokernel of the generating graph of G has an extra automorphism oforder 2, interchanging the sets of vertices coming from elements of orders 3and 5 in G . Twin reduction of this graph is thus not faithful. The cokernelis a graph on 784 vertices with automorphism group C × PΓL(2 ,
11 Above and beyond the hierarchy
This section contains some very brief comments on similar graphs.
Given a subgroup-closed class of graphs C , we can define a graph on G inwhich x and y are joined if h x, y i belongs to C .For C the class of cyclic groups, we obtain the enhanced power graph;and for the class of abelian groups, we obtain the commuting graph.After these, the most natural classes to consider are those of nilpotentand solvable groups; let us denote the corresponding graphs by Nilp( G ) andSol( G ) respectively. 50 Schmidt group is a non-nilpotent group all of whose proper subgroupsare nilpotent. These groups were characterised by Schmidt [91]; see [17] foran accessible account. All are 2-generated.I do not know of a similar characterisation of the non-solvable groupsall of whose proper subgroups are solvable. However, we can conclude thatthey are 2-generated, as follows. Let G be such a group, and S the solvableradical of G (the largest solvable normal subgroup). Then G/S is a non-abelian simple group. (For if
H/S is a minimal normal subgroup of
G/S ,then
H/S is a product of isomorphic simple groups, so H is not solvable, andby minimality H = G .) Now every finite simple group is 2-generated. If wetake two cosets Sg, Sh which generate
G/S , then h g, h i is a subgroup of G which projects onto G/S , and so is non-solvable; by minimality it is equal to G . (In fact we do not need the Classification of Finite Simple Groups here.For clearly G/S is a minimal simple group, and so is covered by Thompson’sclassification of N-groups [100].)It follows that a group G is nilpotent (resp. solvable) if and only if every 2-generated subgroup of G is nilpotent (resp. solvable). For if every 2-generatedsubgroup of G is nilpotent, then G cannot contain a minimal non-nilpotentsubgroup, and so G is nilpotent; similarly for solvability. Proposition 11.1 (a) For any finite group G , we have E (Com( G )) ⊆ E (Nilp( G )) ⊆ E (Sol( G )) .(b) E (Com( G )) = E (Nilp( G )) if and only if all the Sylow subgroups of G are abelian.(c) E (Nilp( G )) = E (Sol( G )) if and only if G is nilpotent.(d) E (Com( G )) = E (Sol( G )) if and only if G is abelian.(e) If G is non-nilpotent, then E (Nilp( G )) ⊆ E (NGen( G )) ; equality holdsif and only if G is a Schmidt group.(f ) If G is non-solvable, then E (Sol( G )) ⊆ E (NGen( G )) ; equality holds ifand only if G is a minimal non-solvable group. Proof (a) The first point is clear from the definition.(b) Suppose that E (Com( G )) = E (Nilp( G )). Then two elements from thesame Sylow subgroup of G generate a nilpotent group; hence they commute.51onversely, if the Sylow subgroups are abelian, then a nilpotent subgroup isthe product of its Sylow subgroups and hence is abelian.(c) Suppose that E (Nilp( G )) = E (Sol( G )). If G is not nilpotent, itcontains a minimal non-snlpotent subgroup, a Schmidt group, which is 2-generated and soluble, hence nilpotent, a contradiction. Conversely, if G isnilpotent, then Nilp( G ) is complete.(d) If Com( G ) and Sol( G ) coincide, then G is nilpotent with abelianSylow subgroups, hence is abelian. The converse is clear.(e), (f) The forward direction in the last two points uses the fact that thesegroups are 2-generated, as remarked above. For if Nilp( G ) and NGen( G ) havethe same edges, then two elements which do not generate G must generatea nilpotent group, and similarly for solvability. (cid:3) Regarding (b), groups with all Sylow subgroups abelian are known, sinceWalter [102] classified the groups with abelian Sylow 2-subgroups. The simplegroups arising here are PSL(2 , q ) with q even or congruent to ± . Question 20
Investigate analogues of the earlier results in this paper in theextended hierarchy of graphs containing Nilp( G ) and Sol( G ).Universality is relatively straightforward, and both cases can be handledtogether. Recall from the proof of Theorem 5.6 that any graph Γ can berepresented as the intersection graph of a linear hypergraph, a family of setswith the property that two sets intersect in 1 point if the correspondingvertices are adjacent, and are disjoiont otherwise. We can add some dummypoints to ensure that all the sets in the collection have the same (prime)cardinality p at least 3. Now if we take cycles whose supports are thesesets, then we see (as there) that the cycles corresponding to adjacent verticesgenerate the alternating group of degree 2 p −
1, while those corresponding tonon-adjacent vertices generate C p × C p . So both nilpotence and solvabilitygraphs of finite groups embed all finite graphs.Connectedness of the complement of the nilpotency graph has been in-vestigated by Lucchini and Nemmi [80].Each of these cases can be stratified: we can define the level- k nilpotenceor solvability graph to have edges { x, y } if h x, y i is nilpotent of class at most k (resp. solvable of derived length at most k ).52ther classes of groups for which the corresponding graphs could be stud-ied, for which the minimal groups not in the class have been considered, in-clude the supersolvable groups (those for which every chief factor is cyclic)and the p -nilpotent groups (groups with normal p -complements). The groupsminimal with respect to not lying in these classes are considered in [16] and[17] respectively.A closely related graph is the Engel graph of a group, defined by Abdol-lahi [3]. Here is a brief account. We define, for each positive integer k , andall x, y ∈ G , the element [ x, k y ] of G to be the left-normed commutator of x and k copies of y ; more formally, • [ x, y ] = [ x, y ] = x − y − xy , • for k >
1, [ x, k y ] = [[ x, k − y ] , y ].Abdollahi defined x and y to be adjacent if [ x, k ] y = 1 and [ y, k x ] = 1 for all k . To fit with the philosophy of this paper, and at Abdollahi’s suggestion,I will redefine it to be the complement of this graph. If we do this then wehave a similar situation to that arising with the power graph. If we definethe directed Engel graph to have an arc from x to y if [ y, k x ] = 1 for some k ,then the Engel graph (that is, the complement of the graph as defined in [3])is the graph in which x and y are joined if there is an arc from one to theother. The directed graph may also have a role to play here.A similar stratification to that for nilpotence and solvability graphs canalso be defined in this case. In particular, the level 1 Engel graph is just thecommuting graph.Zorn [110] showed that, if a finite group G satisfies an Engel identity[ x, k y ] = 1 for all x, y (for some k ), then G is nilpotent; so the finite groupsfor which the directed Engel graph is complete are the same as those forwhich the nilpotency graph is complete. (For infinite groups, this is not true,though the result has been shown in a number of special cases.)So there is a close connection between the Engel graph and the nilpotencygraph. But they are not equal in general. For example, in the group S , thereis an arc of the directed Engel graph from each element of order 3 to eachelement of order 2, but not in the reverse direction. Question 21
What can be said about the relation between the Engel andnilpotency graphs? In particular, in which groups are they equal?53rmed with this knowledge, we return briefly to the nilpotence graph.First, another definition. The upper central series of a group G is the se-quence of subgroups defined by Z ( G ) = { } , Z k +1 ( G ) /Z k ( G ) = Z ( G/Z k ( G )) for k > . The hypercentre of G is the union of these subgroups. Thus, if G is finite, thenthe hypercentre is Z k ( G ), where k is the smallest value such that Z k ( G ) = Z k +1 ( G ). It is clear that we have Z ( G ) = Z ( G ), and subsequent terms arethe subgroups of G that project onto the upper central series of G/Z ( G ).Hence we can give an alternative definition: Z k +1 ( G ) /Z ( G ) = Z k ( G/Z ( G )) . Theorem 11.2
Let G be a finite group. Then the set Z Nilp ( G ) of elementsof G which are joined to all other elements in the nilpotence graph is equalto the hypercentre of G . Proof
Suppose that x is joined to all other elements of G , so that h x, y i is nilpotent for all y ∈ G . In particular, for all y , there exists k such that[ x, k y ] = 1. Baer [14] showed that, in a group with the maximum conditionon subgroups (and in particular a finite group), this implies that x is in thehypercentre of G .For the reverse implication, we have to show that, if x ∈ Z k ( G ), where k is the least value such that Z k ( G ) = Z k +1 ( G ), then for any y ∈ G wehave h x, y i nilpotent. We prove this by induction on k . So assume that itis true with k − k in any group. Then for x ∈ Z k ( G ) we have Z ( G ) x ∈ Z k − ( G/Z ( G )), so Z ( G ) h x, y i /Z ( G ) is nilpotent. Thus Z ( G ) h x, y i is an extension of a central subgroup by a nilpotent group, and so is nilpotent;and so h x, y i is nilpotent, as required. (cid:3) Question 22
What, if anything, can be said for infinite groups?
All the graphs studied so far have the property that two group elementswhich generate the same cyclic subgroup are closed twins. So it would bevery natural to collapse them by factoring out this equivalence relation. Al-ternatively, one could simply remove edges between such pairs, so that they54ecome open twins. Note that the original, the quotient, and the graph withedges removed all have the same cokernel; so, if one of them is a cograph,then they all are.We could put a graph at the bottom of the hierarchy, in which x ∼ y if h x i = h y i ; then the second possibility suggested above fits into our schemeas the difference between this graph and one of the others.I end this section with a general question. Question 23
For which types of graph, and which groups, is the relation ≡ given by x ≡ y if h x i = h y i definable directly from the graph withoutreference to the group?In particular, if the cokernel of the graph is equal to the quotient by theequivalence relation ≡ , this will be true. For which groups is this the case?
12 Intersection graphs
There turns out to be a close connection between certain intersection graphsdefined on G , and some of the graphs in our hierarchy. First I look brieflyat the connection in the abstract, then discuss some particular cases. Let B be a bipartite graph. If it is connected, it has a unique bipartition: takea vertex v ; then the bipartite blocks are the sets of vertices at even (resp. odd)distance from v . If B is not connected, the bipartition is not unique; in fact,there are 2 κ − bipartitions, where κ is the number of connected components,since we can make a bipartite block by choosing a bipartite block in eachcomponent and taking their union. However, I will always assume that thebipartition of B is given, and is part of its structure.The halved graphs arising from B are the graphs Γ and Γ whose vertexset is a bipartite block, two vertices adjacent in the relevant graph if andonly if they lie at distance 2 in B.We call a pair of graphs Γ and Γ a dual pair if there is a bipartite graphB without isolated vertices such that Γ and Γ are the halved graphs of B.I warn that this concept is not the same as the vague notion of dualitywhich informed the name “dual enhanced power graph”. It is however closelyconnected with duality in design theory and geometry, or between a graphand the linear hypergraph (or partial linear space) that we used in Section 5.5.55 roposition 12.1 Let Γ and Γ be a dual pair of graphs. Then Γ is con-nected if and only if Γ is connected. More generally, there is a naturalbijection between connected components of Γ and connected components of Γ with the property that corresponding components have diameters which areeither equal or differ by . Proof
Any vertex of Γ is joined (by an edge of B) to a vertex of Γ , and vice versa , since B has no isolated vertices. Now suppose that two verticesof Γ are joined by a path of length d . Then there is a path of length 2 d in Bjoining them. So a connected component of B is the union of correspondingconnected components in Γ and Γ . Suppose that a component of Γ hasdiameter d . Take two vertices v , v in the corresponding component of Γ .Choose vertices u and u of Γ joined in B to v and v respectively. Thesetwo vertices lie at distance r ≤ d , say; so there is a path of length at most 2 r in B joining them. Thus v and v have distance at most 2 r + 2 in B, whencetheir distance in Γ is at most r + 1, hence at most d + 1. So the diameterof a component of Γ has diameter at most one more than the correspondingcomponent of Γ . Interchanging the roles of the dual pair completes theproof. (cid:3) Question 24
What other relations hold between properties of a dual pairof graphs?If the bipartite graph is semiregular , then a number of properties trans-fer between the corresponding dual pair, especially spectral properties, and(related to this) optimality properties of statistical designs [15].
In order to apply this result, I give a general construction showing that certaingraphs defined on the non-identity elements of a group form dual pairs withcertain intersection graphs of families of subgroups.
Proposition 12.2
Let G be a finite non-cyclic group, and let F be a familyof non-trivial proper subgroups of G with the property that its union is G .Let Γ be the graph defined on the non-identity elements of G by the rule that x is joined to y if and only if there is a subgroup H ∈ F with x, y ∈ H . Then Γ and the intersection graph of F form a dual pair. roof We form the bipartite graph B whose vertex set is ( G \ { } ) ∪ F ,where a group element x = 1 is joined to a subgroup H ∈ F if and only if x ∈ H . We verify the conditions for a dual pair.First, B has no isolated vertices: for each subgroup in F is non-trivial,so contains an element of G \ { } , and every such element is contained in asubgroup in F , since the union of this family is G .Next, two subgroups are joined in the intersection graph if and only iftheir intersection is non-trivial (that is, contains an element of G \ { } ; and,by assumption, two non-trivial elements are adjacent if and only if someelement of F contains both.Note that we have assumed that G is non-cyclic; this in fact follows fromthe fact that it is a union of proper subgroups, since a generator would lie inno proper subgroup. (cid:3) I will consider several cases. I begin with the “classical” case, where thevertices are all the non-trivial proper subgroups of G , joined if two verticesare adjacent. These were first investigated by Cs´ak´any and Poll´ak, whoconsidered non-simple groups; they determined the groups for which theintersection graph is connected and showed that, in these cases, its diameteris at most 4. For simple groups, Shen [95] showed that the graph is connectedand asked for an upper bound; Herzog et al. [63] gave a bound of 64, whichwas improved to 28 by Ma [81], and to the best possible 5 by Freedman [49],who showed that the upper bound is attained only by the Baby Monster andsome unitary groups (it is not currently known exactly which). Proposition 12.3
Let G be a non-cyclic finite group. Then the inducedsubgraph of the non-generating graph of G on non-identity elements and theintersection graph of G form a dual pair. Proof
Take F to be the family of all non-trivial proper subgroups of G . (cid:3) So the reduced non-generating graph of a non-abelian finite simple grouphas diameter at most 6; this bound can be reduced to 5, and possibly to 4,perhaps with specified exceptions (Saul Freedman, personal communication).Now we turn to the commuting graph.57 roposition 12.4
Let G be a finite group with Z ( G ) = 1 . Then the reducedcommuting graph of G (on the vertex set G \ { } ) and the intersection graphof non-trivial abelian subgroups of G form a dual pair. Proof
The condition Z ( G ) = 1 ensures that the reduced commuting graphdoes have vertex set G \ { } , and also implies that G is not cyclic. Take F to be the family of all non-trivial abelian subgroups of G (all are propersubgroups since G is not abelian). Two elements are joined in the reducedcommuting graph if and only if the group they generate is abelian. (cid:3) Corollary 12.5
For a finite group G with Z ( G ) = { } , the following fourconditions are equivalent:(a) the Gruenberg–Kegel graph of G is connected;(b) the reduced commuting graph of G is connected;(c) the intersection graph of non-trivial abelian subgroups of G is connected;(d) the intersection graph of maximal abelian subgroups of G is connected. Proof
The equivalence of (a) and (b) comes from Theorem 9.2, and thatof (b) and (c) from Proposition 12.1. For the equivalence of (c) and (d),note that any non-trivial abelian subgroup is contained in a maximal abeliansubgroup, to which it is joined, so (d) implies (c). The converse holds becauseany path in the intersection graph of non-trivial abelian subgroups can belifted to a path in the intersection graph of maximal abelian subgroups. (cid:3)
Proposition 12.6
Let G be a group which is not cyclic or generalised quater-nion. Then the induced subgraph of the enhanced power graph of G on theset of non-identity elements and the intersection graph of non-trivial cyclicsubgroups of G form a dual pair. In fact the theorem applies also to generalised quaternion groups; but forthese, both the reduced enhanced power graph and the intersection graphare connected for the trivial reason that they contain a vertex joined to allothers.
Proof
We take F to be the family of non-trivial cyclic subgroups of G . (cid:3) G ) and collapse the equivalenceclasses of the relation ≡ , where x ≡ y if h x i = h y i , we obtain the intersectiongraph of non-trivial cyclic subgroups of G . (This is probably why it wascalled the “intersection graph” in [40].)A study of intersection graphs of cyclic subgroups has been published byRajkumar and Devi [88].
13 More general graphs
When we think about graphs on groups, we want there to be some connectionbetween the graph and the group. This connection is mostly expressed interms of invariance of the graph under something, either right translationsor automorphisms of the group. The first gives rise to Cayley graphs, asdiscussed briefly in Section 1.2.So the focus here is on graphs on a group G invariant under the automor-phism group Aut( G ) of G . We have seen that all graphs in the hierarchy dosatisfy this condition.There are several ways we could approach the general case. • Any graph invariant under Aut( G ) is a union of orbital graphs forAut( G ). • We could define the adjacency in the graph by a first-order formulawith two free variables. • We could define adjacency by some more recondite group-theoreticproperty.We will see examples below.However, it matters whether we are defining the graph on a single group,or defining it on the class of all groups.
If we are given a group G , and can compute Aut( G ), then the first procedure(taking unions of orbital graphs) obviously gives all orbital (di)graphs for G . Theorem 13.1
Given a group G , for every Aut( G ) -invariant graph, thereis a formula φ in the first-order language of groups such that x ∼ y if andonly if G | = φ ( x, y ) . roof By the so-called Ryll-Nardzewski Theorem, proved also by Engelerand by Svenonius (see [65]), G is oligomorphic, so the G -orbits on n -tuplesare n -types over G , that is, maximal sets of n -variable formulae consistentwith the theory of G ; but all types are principal, so each is given by a singleformula. (cid:3) Question 25
Given G , is there a bound for the complexity of the formulaedefining orbital graphs for Aut( G ) acting on G (for example, for the alterna-tion of quantifiers)?Clearly the commuting graph can be defined by the quantifier-free formula xy = yx . If G is an elementary abelian 2-group, there are only three (non-diagonal) orbital graphs, defined by the formulae ( x = 1) ∧ ( y = 1), ( x =1) ∧ ( y = 1), and ( x = 1) ∧ ( y = 1) ∧ ( x = y ) respectively. As we have seen, the commuting graph is defined uniformly for all groups bythe quantifier-free formula xy = yx .It seems unlikely that the other graphs listed earlier have uniform first-order definitions. The statement h x, y i = G seems to require quantificationeither over words in x, y or over subsets of G , and so to need some version ofhigher-order logic for its definition.For example, suppose that there is a formula φ ( x, y ) which, in any finitegroup, specifies that x and y are joined in the power graph. Taking C n with n even, with x a generator and y of order 2, the formula is always satisfied.So it should hold in an ultraproduct of such groups (see [19]). But in theultraproduct, x has infinite order and y has order 2, so y cannot be a powerof x . If we are given a specific group G and know its automorphism group, thenconstructing all the orbital graphs is a simple polynomial-time procedure.If we are given G and don’t know (and maybe are trying to find out about)its automorphism group, then clearly some indication of which first-orderformulae need to be considered would be helpful. Maybe, given g, h ∈ G , thetype of ( g, h ) (the set of φ ( x, y ) such that G | = φ ( g, h )) could be described,and a formula generating the type found.60nother situation that might arise would be that we are given one ormore graphs defined on general groups and are interested to know for whichgroups they have some property, e.g. two graphs equal. If we had first-orderdescriptions of the graphs, we would just be looking for models of some first-order sentence.
14 Infinite groups
The definitions of the graphs in the hierarchy (with the exception of the deepcommuting graph), and the inclusions among them, work without change forinfinite groups. I will simply mention a few highlights here, as there is littlein the way of general theory.
Let p be a prime number. The Pr¨ufer group G = C p ∞ is the group of rationalnumbers with p -power denominators mod 1, or the multiplicative group of p -power roots of unity. Every element of the group has p -power order, andthe group has a unique subgroup of order p n for any n . It follows that thepower graph of G is a countable complete graph, independent of the choiceof prime.The directed power graph does determine the prime, since the class ofelements immediately above the identity has size p −
1. This shows that thepower graph does not determine the directed power graph for infinite groupsin general.However, the implication does hold for torsion-free groups. This wasshown by Zahirovi´c [108]; a preliminary result appears in [34]. In fact, thehypotheses in Zahirovi´c’s result are weaker; I refer to the paper for details,which show the important role played by the Pr¨ufer groups in this problem.
Perhaps the most striking result on the commuting graph of an infinite groupis the following, due to Bernhard Neumann (answering a question of PaulErd˝os):
Theorem 14.1
Let G be an infinite group. Then the following are equiva-lent: a) Com( G ) has no infinite coclique;(b) there is a finite upper bound on the size of cocliques in Com( G ) ;(c) Z ( G ) has finite index in G . I have stated Neumann’s result like this for comparison with what follows.He proved that (a) implies (b) and (c). Now (b) implies (a) is trivial, and (c)implies (b) because if (c) holds, then G is a finite union of abelian subgroups(since h Z ( G ) , g i is abelian for all g ∈ G ), and a coclique in Com( G ) cancontain at most one vertex from each subgroup.What about the power graph or enhanced power graph?Certainly, if either of these graphs has no infinite coclique, then neitherdoes Com( G ); so Z ( G ) has finite index in G . But consider the group G = C p ∞ × C q ∞ , where p and q are distinct primes. It is easy to show that Pow( G )has no infinite coclique; but, if a n has order p n and b n has order q n , then { a n b , a n − b , . . . , a b n − , a b n } is a coclique of size n + 1, for any n .However, if Pow( G ) has no infinite coclique, then G is the union of finitelymany abelian subgroups. So we first ask, which abelian groups have no in-finite coclique? Such a group must be a torsion group; for, if a were anelement of infinite order, then { a p : p prime } is an infinite coclique in thepower graph. There can only be finitely many primes such that G containselement of order p . So G is the direct sum of its finitely many Sylow sub-groups. Moreover, the Sylow subgroups must have finite rank. So we canconclude: Theorem 14.2
Let G be an infinite group. Then the following are equiva-lent:(a) Pow( G ) has no infinite coclique;(b) Z ( G ) has finite index in G and is a direct sum of finitely many p -torsionsubgroups of finite rank, for primes p . So G is locally finite, a result of Shitov [96].If we make the stronger hypothesis that the size of cocliques is bounded,then we can strengthen the conclusion to assert that all but one of the Sylowsubgroups of Z ( G ) is finite.For the enhanced power graph, Abdollahi and Hassanabadi [5] provedthat the analogue of Neumann’s Theorem does hold:62 heorem 14.3 Let G be an infinite group. Then the following are equiva-lent:(a) EPow( G ) has no infinite coclique;(b) there is a finite upper bound for the size of cocliques in EPow( G ) ;(c) Z EPow ( G ) has finite index in G . (Recall that Z EPow ( G ) is the cyclicizer of G .) Another significant body of work on power graphs concerns the clique pa-rameters. Here are some striking results, which appear in [1, 35, 96].
Theorem 14.4
Any infinite group has clique number and chromatic numberat most countable.
Theorem 14.5
For an infinite group G , the following conditions are equiv-alent:(a) Pow( G ) has finite clique number;(b) Pow( G ) has finite chromatic number;(c) EPow( G ) has finite clique number;(d) EPow( G ) has finite chromatic number;(e) G is a torsion group with finite exponent. Proof
The power graph of an infinite cyclic group h g i contains an infiniteclique { g n : n ≥ } . So a group satisfying any of the first four conditionsis a torsion group. Now the results are proved just as for finite groups inSection 4. (cid:3) The cited papers contain other miscellaneous results.63
The ideas behind some of these graphs can be extended to other algebraicstructures.A magma is a set with a binary operation. (The term groupoid is some-times used, but I will avoid this since it is also used for a category in whichevery morphism is invertible.) Beyond groups, the two classes of magmasmost studied are semigroups (satisfying the associative law) and quasigroups (in which left and right division are well-defined), and in particular monoids and loops (semigroups, resp. quasigroups, with identity elements).Clearly the definition of commuting graph makes sense in any magma. Forthe power graph and its relatives, it is necessary to make sense of powers of anelement. One can define left powers inductively by a = a and a n +1 = a ◦ a n for n ≥
1. (This is the approach adopted in [101].) An alternative is torestrict to power-associative magmas , those in which the product of n termseach equal to an element a is independent of the bracketing used to evaluateit. Now, just as for groups, we have: Proposition 15.1
In a power-associative magma, the directed power graphis a partial preorder, and so the power graph is the comparability graph of apartial order.
Question 26
For which magmas, or quasigroups, is the power graph (de-fined using left powers) a comparability graph of a partial order, or a perfectgraph?The power graph of a semigroup was defined early in their study of powergraphs, see [72]).The commuting graphs of semigroups are considered by Ara´ujo et al. [11],who pose a number of questions about them.As for groups, the power graph of a semigroup is a spanning subgraphof its commuting graph. The enhanced power graph could be defined forany semigroup; to my knowledge this has not been studied. It is not clearwhether the definition of the deep commuting graph could be adapted forsemigroups.Commuting graphs of semigroups are universal (and power graphs areuniversal for comparability graphs of partial orders), since these statementshold for groups. 64he intersection graph of the subsemigroups of a semigroup had beenstudied much earlier: Bos´ak [25] raised the question of its connectedness in1963, and the question was soon resolved by Lin [77] and Pondˇeliˇcek [87]:for any finite semigroup, this graph is connected with diameter at most 3.These results preceded the investigation of the intersection graph for groups,mentioned earlier. (The intersection graphs of semigroups are not comparablewith the intersection graphs of groups, since any subgroup of a group containsthe identity, so adjacency requires their intersection to be non-trivial.)The commuting graph (and its complement), the intersection graph ofcyclic subgroups, and the power graph have also been studied for quasi-groups and loops, especially for the classes of Moufang and Bol loops: see,for example, [9, 61, 62, 101]. Moufang loops form a class of loops which isperhaps closest to groups: a Moufang loop is a loop satisfying the identity z ( x ( zy )) = (( zx ) z ) y (a weakening of the associative law). In particular, a2-generated subloop of a Moufang loop is associative; so a Moufang loop ispower-associative, and its power graph is the comparability graph of a partialorder.A question raised some time ago but to my knowledge not yet answeredis: Question 27
If the power graphs of two finite Moufang loops are isomor-phic, are their directed power graphs isomorphic?This question has been investigated by Nick Britten, who has found nocounterexamples among the Moufang loops in the LOOPS package [84] for
GAP (Michael Kinyon, personal communication).Going beyond a single binary operation, we reach the class of rings. Forthese, the zero-divisor graph of a ring was introduced by Beck [18] in 1988:the vertices are the ring elements, with a and b joined whenever ab = 0. Ifthe ring is commutative, the graph is undirected. Another graph associatedwith a ring is the unit graph , in which a and b are joined whenever a + b isa unit [13]. References [1] G. Aalipour, S. Akbari, P. J. Cameron, R. Nikandish and F. Shaveisi,On the structure of the power graph and the enhanced power graph ofa group,
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