Mary Agnes SERVATIUS Izomorfni Cayleyevi grafi nad neizomorfnimi grupami (Isomorphic Cayley Graphs on Non-Isomorphic Groups)

Transcription

1 UNIVERZA NA PRIMORSKEM FAKULTETA ZA MATEMATIKO, NARAVOSLOVJE IN INFORMACIJSKE TEHNOLOGIJE Matematične znanosti Študijski program 2. stopnje Mary Agnes SERVATIUS Izomorfni Cayleyevi grafi nad neizomorfnimi grupami (Isomorphic Cayley Graphs on Non-Isomorphic Groups) Magistrsko delo (Master s Thesis) Mentor: izr. prof. dr. István Kovács Koper, 2014

2 Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 Ključna dokumentacijska informacija ii Ime in PRIIMEK: Mary Agnes SERVATIUS Naslov magistrske naloge: Izomorfni Cayleyevi grafi nad neizomorfnimi grupami Kraj: Koper Leto: 2014 Število listov: 56 Število slik: 27 Število referenc: 18 Mentor: izr. prof. dr. István Kovács UDK: Ključne besede: Cayleyjev graf, cikličen graf, isomorfnost grafov, grupa avtomorfizmov grafov, leksikografičen produkt grafov, končna abelova grupa, permutacijska grupa, venčni produkt permutacijskih grup, ločno tranzitiven graf. Math. Subj. Class. (2010): 05C25 20B25 20B25 05C60 Izvleček V magistrskem delu obravnavamo Cayleyeve grafe in grupe. Poseben poudarek je namenjen problemu Cayleyevih izomorfizmov. Poglavje 1 vsebuje uvod v glavni izrek, ki poda potrebni in zadostni pogoj za izomorfnost dveh Cayleyevih digrafov X 1 = cay(g 1, S 1 ) in X 2 = cay(g 2, S 2 ), kjer sta G 1 in G 2 neizomorfni abelski 2- grupi, digrafa X 1 in X 2 pa imata regularno ciklično grupo avtomorfizmov. Omenjeni rezultat je razširitev rezultata Morrisove (glej J. Graph Theory 3 (1999), ) v zvezi s p-grupami G i, kjer je p liho praštevilo. Poglavje 2 vsebuje osnovne definicije in potrebno predpripravo za kasnejša poglavja. V 3. poglavju so obravnavani izbrani primeri. V 4. poglavju je podan dokaz glavnega izreka.

3 Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 iii Key words documentation Name and SURNAME: Mary Agnes SERVATIUS Title of Masters degree: Isomorphic Cayley Graphs on Non-isomorphic Groups Place: Koper Year: 2014 Number of pages: 56 Number of figures: 27 Number of references: 18 Mentor: Assoc. Prof. István Kovács, PhD UDK: Key words: Cayley graph, cyclic graph, isomorphic graphs, automorphism groups of graphs, lexicographic products of graphs, elementary abelian groups, permutation groups, wreath product, arc transitive graph Math. Subj. Class. (2010): 05C25 20B25 20B25 05C60 Abstract In this thesis we explore Cayley graphs and groups, in particular, we are concerned with the Cayley isomorphism problem. Chapter 1 contains a high level introduction to the main theorem, in which a necessary and sufficient condition is given for two Cayley digraphs X 1 = Cay(G 1, S 1 ) and X 2 = Cay(G 2, S 2 ) to be isomorphic, where G 1 and G 2 are non-isomorphic abelian 2-groups, and the digraphs X 1 and X 2 have a regular cyclic group of automorphisms. This result extends that of Morris (see J. Graph Theory 3 (1999), ) concerning p-groups G i, where p is an odd prime. Chapter 2 introduces some preliminaries, and Chapter 3 provides a selection of examples. Chapter 4 contains a proof of the main theorem.

4 Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 iv Contents 1 Introduction 1 2 Preliminaries Graphs Basic definitions Groups Basic definitions Automorphism groups Permutation groups and group actions Imprimitivity block systems Wreath products of groups Cayley graphs Arc transitive Cayley digraphs on Cyclic groups CI-Graphs Non-isomorphic groups with isomorphic Cayley graphs Infinite graphs and free groups Graphs of the Platonic solids Proof of the Main Theorem W-subgroups of abelian groups Proof of Theorem Conclusions 48

5 Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 v Bibliography 49

6 Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 vi Acknowledgements Thanks are due to Dragan Marusič for this opportunity, to my mentor István Kovács for his patience, and to my father, Herman Servatius, who taught me everything I know. Thank you Papa, for letting me peek at the vast explored and unexplored territories of mathematics, for going on random walks with me on objects in space, and most of all for distracting me in my most difficult times with your apparently endless supply of mathematical treats. It does not escape my notice that the time, energy and money spent on this project vastly outweigh its value. Many years ago, when I was much too young and naive to be suspicious of a grand promise, I signed a contract which effectively forced me to work for years at a job to which I was completely unqualified for less than nothing. My anger over this situation is the only thing which shields me from my grief and humiliation. I am not sure if I will ever completely recover. Work on this thesis was generously supported by grant TI from the Trowbridge Institute, Massachusetts, USA. My maiden name is Mary Agnes Franziska Sophie Servatius, and my married name is Mary Milanič. The name on the cover of this thesis does not reflect the legal status of my name in the countries of which I am a citizen, although it is being used to refer to me.

7 Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, Chapter 1 Introduction A necessary and sufficient condition is given for two Cayley digraphs X 1 = Cay(G 1, S 1 ) and X 2 = Cay(G 2, S 2 ) to be isomorphic, where the groups G i are nonisomorphic abelian 2-groups, and the digraphs X i have a regular cyclic group of automorphisms. Our result extends that of Morris (see J. Graph Theory 3 (1999), ) concerning p-groups G i, where p is an odd prime. Let G be a finite group with identity element e. A Cayley digraph Cay(G, S) on G generated by a connection set S G, e / S, is a digraph with the vertex set G and the arc set {(x, y) x 1 y S}. In this paper we consider the problem of finding efficient necessary and sufficient conditions for two Cayley digraphs Cay(G 1, S 1 ) and Cay(G 2, S 2 ) to be isomorphic, such that the groups G i are nonisomorphic. Here we restrict to the special case that G i are p-groups. For an overview on the general problem we refer to the survey paper by Li [12]. The simplest case is that the groups G i have order p 2, and this has been characterized by Joseph in [10]. Joseph s result was generalized to G i = p n by Morris in [16] as follows. Recall first that, for two digraphs X 1 and X 2, the wreath product X 1 X 2 is the digraph with vertex set V (X 1 ) V (X 2 ), and an arc from (x 1, x 2 ) to (y 1, y 2 ) if and only if (x 1, y 1 ) is an arc of X 1, or x 1 = y 1 and (x 2, y 2 ) is an arc of X 2. We remark that the wreath product Aut(X 1 ) Aut(X 2 ) acts as a group of automorphisms of X 1 X 2, explaining also the term wreath product. Here we follow the convention that Aut(X 1 ) Aut(X 2 ) has active group Aut(X 1 ) and passive group Aut(X 2 ), i.e.,

8 Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, the order Aut(X 1 ) Aut(X 2 ) = Aut(X 1 ) Aut(X 2 ) X1. Second, the set of all abelian groups of a fixed prime-power order p n is partially ordered by the relation po, where for two abelian groups G and H of order p n, G po H if and only if there is chain H 1 < H 2 < < H m = H of subgroups of H such that 1. H 1, H 2 /H 1,..., H m /H m 1 are all cyclic, and 2. G = H 1 H 2 /H 1 H m /H m 1. The main result of Morris is the following (see [16, Theorem 1.1]). Theorem Let X = Cay(G, S) be a Cayley digraph on an abelian group G of order p n, where p is an odd prime. Then the following are equivalent: (1) The digraph X is isomorphic to a Cayley digraph on both Z p n and H, where H is an abelian group with H = p n. (2) There exists a chain of subgroups G 1 < < G m 1 in G such that (i) G 1, G 2 /G 1,..., G/G m 1 are cyclic groups, (ii) G 1 G 2 /G 1 G/G m 1 po H, (iii) for each 1 i m 1, S \ G i is a union of G i -cosets. (3) There exist Cayley digraphs X 1,..., X m on cyclic p-groups H 1,..., H m such that H 1 H m po H and X = X m X 1. Moreover, Theorem with the assumption G = H was shown to be true for any (not necessarily abelian) group G of order p n (see [16, Theorem 6.1]), as well as when p = 2 and H is an elementary abelian group of order 2 n (see [16, Theorem 5.1]). Our goal in this thesis is to extend Theorem to the case when p = 2, that is, to show the following theorem.

9 Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, Theorem Let X = Cay(G, S) be a Cayley digraph on an abelian group G of order 2 n. Then the following are equivalent: (1) The digraph X is isomorphic to a Cayley digraph on both Z 2 n and H, where H is an abelian group with H = 2 n. (2) There exists a chain of subgroups G 1 < < G m 1 in G such that (i) G 1, G 2 /G 1,..., G/G m 1 are cyclic groups, (ii) G 1 G 2 /G 1 G/G m 1 po H, (iii) for each 1 i m 1, S \ G i is a union of G i -cosets. (3) There exist Cayley digraphs X 1,..., X m on cyclic 2-groups H 1,..., H m such that H 1 H m po H and X = X m X 1. Notice that the aforementioned nonabelian version of Theorem does not hold for p = 2. Consider the cycle C 2 n on 2 n vertices. Although the graph C 2 n is not decomposed nontrivially into wreath products of smaller graphs, it admits a regular group of automorphisms isomorphic to Z 2 n, and another one isomorphic to the dihedral group D 2 n. We remark that our result has been published in: J. Graph Theory 70 (2012), An outline of the paper is as follows: Chapter 2 contains a very basic and friendly introduction to graphs and groups, though focusing particular attention on Cayley graphs, permutation groups, automorphism groups and wreath products. We begin chapter 3 with some nice examples of isomorphic Cayley graphs on non-isomorphic groups. We continue with some interesting results from the literature, and end the chapter as well as this thesis with the proof of our main theorem.

10 Chapter 2 Preliminaries In this chapter we will review the definitions and results from graph and group theory respectively that will be needed in the thesis. For a deeper look into graph theory, see [2]. For more information about permutation groups, see [4] and for more about algebra, I would recommend [7] and [8]. For those desiring a more intuitive approach to learning group theory, I highly recommend [3]. 2.1 Graphs Basic definitions A simple graph, also called an undirected graph X = (V, E) consists of a vertex set V = V (X) and an edge set E = E(X), where each edge is an unordered pair {v 1, v 2 } of distinct vertices, v 1, v 2 V. The graphs in Figure 2.1 are all simple and undirected graphs. Two vertices which are contained in an edge are said to be adjacent to one another and incident to that edge. Two edges with a vertex in common are also said to be incident to one another. If X = (V, E) and X = (V, E ) are graphs and V V and E E, then X is said to be a subgraph of X, and we write X X. If X X then any pair of vertices of V which are adjacent in X are adjacent as well in X. If, moreover, any pair v 1 and v 2 of vertices in V which are adjacent in X are also adjacent in X, then 4

11 Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, the subgraph X is said to be full or induced. The degree of a vertex is the number of edges to which it is incident. The sum of the degrees over all vertices in a graph is equal to twice the number of edges in the graph. A graph is called r-regular if every vertex has degree r. Two very important graphs are cycles and paths, mostly because of their roles as subgraphs. The path P n on n vertices is the graph with V (P n ) = {1,..., n} and E(P n ) = {{1, 2}, {2, 3},..., {n 1, n}}. The vertices 1 and n are the only vertices of degree 1 in the path, and are said to be end vertices, see Figure 2.1. Similarly, the cycle C n on n vertices, n 3, is a the graph with V (C n ) = {1,..., n} and E(C n ) = {{1, 2}, {2, 3},..., {n 1, n}, {n, 1}}. Cycles are 2-regular graphs, in fact, any 2-regular graph is either a cycle or a disjoint union of cycles. Figure 2.1: Some simple graphs. A graph X is connected if each pair of vertices are the end vertices of some subgraph which is a path. We say simply that the vertices are joined by a path. More generally, digraph X consists of a vertex set V (X) and a set E(X) of arcs (sometimes denoted A(X)), or ordered pairs of distinct vertices. In general, an arc (x, y) is represented in figures by an arrow from x to y. If both (x, y) and the opposite pointing arc (y, x) are in E(X), the two arcs can be represented by a single

12 Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, undirected edge. Even more generally, in a multigraph the edge set consists of ordered pairs of vertices, such that each arc is given a multiplicity. Thus, both loops, or edges of the form (x, x), and multiple edges between pairs of vertices are permitted. Two (di)graphs X and Y are said to be isomorphic if there exists a bijection φ : V (X) V (Y ) between the vertex sets such (x, y) E(X) if and only if (φ(x), φ(y)) E(Y ). For isomorphic (di)graphs X and Y we use X = Y. In particular, an automorphism is an isomorphism of a graph onto itself. The automorphisms of a graph form a group, the automorphism group discussed in the next section. In this thesis, we will only consider simple graphs and digraphs, so multiple edges or loops will not be allowed, however disconnected graphs are allowed. 2.2 Groups Basic definitions A group is a set G together with a binary operation : G G G, written (g 1, g 2 ) = g 1 g 2, which is associative, g 1 (g 2 g 3 ) = (g 1 g 2 ) g 3, is such that there is an identity element e G with e g = g e = g for all g G, and is such that each element g G has an inverse element g G such that g g = g g = e. It is common practice that if the multiplication is also commutative, g 1 g 2 = g 2 g 1 for all g 1, g 2 G, then the operation is termed an addition, written with +, the identity element is denoted by 0, the inverse element of g is denoted by g, and the group is said to be abelian. Otherwise, in the case of a not-necessarily abelian group, the operation is regarded as multiplication, the identity is denoted by 1, and the inverse of g is written as g 1. A group homomorphism is a function f : G 1 G 2 between the underlying sets of two groups G 1 and G 2 which respects multiplication, inverses, and the identity, i.e. f(g h) = f(g) f(h); f(g 1 ) = f(g) 1 ; f(1) = f(1). If the function is bijective, the mapping is an isomorphism. In general, the set of

13 Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, elements K f = {g G f(g) = 1}, the kernel of f is a normal subgroup of G. For isomorphic groups G 1 and G 2 we write G 1 = G2. A group G is said to be generated by a subset S G if every element g G can be expressed in terms of the elements of S, their inverses, and the group multiplication. The simplest examples are the cyclic groups which are generated by a single element. Cyclic groups are necessarily abelian, so were let 1 denote the generator, then there are two cases. If there are an infinite number of elements the group is isomorphic to the additive group Z of integers. If there are only finitely many elements then the cyclic group is isomorphic to Z n, the additive group of integers modulo n. The cardinality of the set G is called the order of the group. A (proper) subset G 1 G of a group G which is closed under the operation and the taking of inverses is said to be a (proper) subgroup of G, and G is said to be an overgroup of G 1, and we write G 1 G. For example, the cyclic group Z 15 has four subgroups, the trivial group {0}, 5Z 15 = {0, 5, 10} = Z 3, 3Z 15 = {0, 3, 6, 9, 12} = Z 5, and Z 15 itself. Every element g of a group G generates a cyclic subgroup of G, and order of that cyclic subgroup is called the order of the element g. Given a group G, a subgroup H G, and an element g G, the set gh = {gh g H} is a left coset of H. Clearly gh = H and it is easy to show that the left cosets of G by H partition the set G, establishing Lagrange s Theorem, that the order of a subgroup divides the order of a group. In particular, the order of any element of G divides the order of G. Similarly, one can define right cosets Hg = {hg h H}, and if gh = Hg for all g G, then the subgroup is said to be normal. If H is a normal subgroup then (gh)(g H) = g(hg )H = (gg )(HH) = (gg )H so the multiplication of cosets is well-defined giving the set of cosets, G/H a group structure induced from the multiplication of G. The group of cosets G/H is called a factor group. Given two groups G 1 and G 2, their cartesian product is the group G 1 G 2 whose elements are the set of ordered pairs {(g 1, g 2 ) g 1 G 1, g 2 G 2 }

14 Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, with multiplication defined in each coordinate separately, (g 1, g 2 )(g 1, g 2) = (g 1 g 1, g 2 g 2). Clearly if G 1 and G 2 are both abelian, then G 1 G 2 is abelian. For example, Z 3 Z 5 is abelian and has fifteen elements and is generated by the set {(1, 0), (0, 1)}. It is, in fact, cyclic, generated by {(1, 1)}, and Z 15 = Z3 Z 5. In general, given two primes p and q, Z pq = Zp Z q. On the other hand Z 9 = Z3 Z 3, since Z 3 Z 3 only has elements of order 1 and 3. In general we have the following. Theorem (Fundamental Theorem of Abelian Groups). Every finitely generated abelian group G is isomorphic to a product of cyclic groups of infinite or prime power order G = Z Z Z ɛ p 1 1 Z ɛ p k k = Z n Z ɛ p 1 1 Z ɛ p k k where the ɛ i are positive integers and the p i are not-necessarily distinct primes. Moreover, the expression is unique provided that p i p i+1 and ɛ i ɛ i+1 for those exponents associated with the same prime. An elementary abelian p-group is a finitely generated abelian group all of whose elements are of order p, so only one prime p occurs in its expression (see Theorem 2.2.1). Of particular interest in this thesis are the 2-groups, all of whose elements are of 2-power order Automorphism groups An important source of groups is the set of automorphisms of a structure. Specifically, given a graph X = (V, E), the set of all automorphisms of X clearly forms a group under composition. It is significant that the converse of this observation

15 Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, is also true, as expressed in Frucht s Theorem which says that every group is the automorphism group of some graph. It is illustrative to attempt to find a graph X such that Aut(X) = G for some G. For the trivial group, we can take, for instance, the one vertex graph, otherwise known as the trivial graph, but any graph with no non-trivial automorphisms is sufficient. A stronger version of Frucht s theorem proved a decade later states that any group is the automorphism group of a cubic graph, and in fact the smallest cubic graph with a trivial automorphism group was named after Frucht. The Frucht graph is shown in Figure 2.2 Figure 2.2: The Frucht graph For a slightly more substantial example, let us take a look at the cyclic groups. Because of the simple structure of such groups, it is easy to find graphs whose automorphism groups have a subgroup isomorphic to a cyclic group. In particular, any graph exhibiting rotational symmetry has an automorphism group with such a subgroup. For example, an automorphism of the cycle graph C n n n-2 3 n is determined by the images of the vertices 0 and 1. There are n possible images of 0, and each is realized by one of the rotations ρ k (i) = i + k, and setting τ(i) = i,

16 Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, the n reflections ρ k τ complete the 2n automorphisms found the automorphism group of the n-cycle graph, defining the dihedral group D 2n. The dihedral group for n > 2 is the simplest example of a non-abelian group, and the subgroup of the dihedral group induced by rotations is always a cyclic group. In order to find a graph whose automorphism group is isomorphic to a cyclic group Z n with n > 2 we have to find a way to allow rotations while preventing reflections or other unwanted automorphisms. To do this we need more vertices, at least twice as many in fact. The smallest graph whose automorphism group is Z 3 is shown in Figure 2.3 and had 9 vertices. Figure 2.3: The smallest cyclic group graph One way to easily find (not necessarily the smallest) graphs with the automorphism group Z n is described by Lovàsz in his selection of combinatorial problems and exercises [15]. Construct a cycle of length 3n and append paths of length 1 and 2 to every third vertex. Because any reflection can not map paths of the same length onto each other, rotations are the only automorphisms, thus yielding the desired automorphism group. The first three such graphs are shown in Figure 2.4. Figure 2.4: Graphs with the automorphism groups Z 3, Z 4, and Z 5. To end this section, we will construct connected graphs whose automorphism

17 Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, groups are the elementary abelian groups Z2 n. It is easy to check that Aut(K 2 ) = Z 2. For Z2 2 we might try to take two copies of K 2, but this graph has the unwanted automorphism of switching the copies, and has an isomorphism group isomorphic to the dihedral group of order 4. This is because the complement of two copies of K 2 is precisely C 4, and since an automorphism maps edges to edges and nonedges to nonedges, Aut(G) = Aut(G). To prevent the appearance of such unwanted automorphisms, we need to find non-isomorphic graphs each with the same automorphism group, and a simple class of such graphs are the paths. Thus Z2 n is the automorphism group of a (disconnected) graph consisting of n paths of different lengths. The complement of a disconnected graph is connected, and the first three graphs constructed in this way are shown in Figure 2.5. Figure 2.5: The automorphism groups of these graphs are Z 2, Z 2 2 and Z 3 2 respectively Permutation groups and group actions Let S be a finite set. Then Sym(S) denotes the set of all bijections, or permutations of S. The set Sym(S) is a group under composition, with cardinality Sym(S) = S!. Any subgroup of Sym(S) is also called a permutation group, so in particular the automorphism group of a graph is a permutation group and Cayley s Theorem implies that every finite group is isomorphic to a permutation group. More generally any group homomorphism φ : G Sym(S) establishes a correspondence between the elements of G and a set of permutations of S, and we say that G acts on S. Often the notation for the homomorphism φ is suppressed and we write φ(g)(s) = gs. If the homomorphism φ is one-to-one, so only the identity element of g acts as the identity on S, the action is said to be faithful. The action is

18 Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, transitive if for any pair of elements x, y S there exists a g G such that gx = y. An action is free if the only element of G which fixes any element of s is the identity, or, equivalently, if gs = hs for some s S implies g = h. If the group G acts on S, then given any element s S, the orbit of s under G is the set s G = {gs g G} and the stabilizer of s is the set G s = {g G gs = s. For each element s S it is easy to see that G = s G G s. One important example of a group action is the action of a group G on the set of (left) cosets G/H of G with respect to some subgroup H, defined by g(xh) = (gx)h. If H is the trivial subgroup consisting only the identity element, this action is called the left regular action. As another example, consider the group of symmetries of the cube. The group acts transitively but not freely on the set of vertices, since, for instance, there is 120 rotation about any pair of antipodal vertices which fixes both of them, but no other vertices. This group acts transitively as well on the set edges, and the set of faces. There is a transitive action as well on the set of arcs, where an arc is a pair (v, e) consisting of one vertex, v and one edge, e which is incident to v. This action of the symmetry group on the arcs is also not free, since the mirror of a diagonal reflection across a plane contains four arcs, each of which are fixed, while all other arcs are not fixed. The symmetries of the cube also acts on the set of flags, that is, the set of triples (v, e, f) where v is a vertex, e and edge incident to v, and f is a face containing e, which can be visualized as a triangular section of each face, see Figure 2.6. This action is both transitive and free. Note that the group of rotational symmetries of the cube acts freely but not transitively on the set of flags, since there is no rotation which will map a white triangle onto a green triangle. An action which is both free and transitive is called regular. A regular action makes it possible to identify in a very natural way the objects of the set S with the elements of the group G. Specifically, arbitrarily choose one element x of S and label it with the identity element 1 of G. Then label each of the other elements of S with the unique element g of G which maps the element labeled 1 to it, so if g(x) = y,

19 Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, Figure 2.6: The symmetries of the cube act on the vertices, the edges, the faces, the arcs, and the flags of the cube. label y with g. Now g (y) = g (g(x)) = (g g)x, so the action of any element g is to take the element labeled g to the element labeled g g. Thus the labeling, in other words the choice of x, identifies the given action with the left regular action defined above Imprimitivity block systems Let G be a group acting transitively on a set Ω, and let B be an imprimitivity block system (called block system for short) of G. Denote by G B the corresponding kernel, that is, G B = {g G g(b) = B for all B B}. Furthermore, denote the permutation group of B induced by the action of G on B by G B. For a block B, denote by G {B} the set-wise stabilizer of B in G, and for a subgroup H G {B}, denote the restriction of H to B by H B. For an arbitrary group G and g G, let g L be the permutation in Sym(G) defined by g(x) = gx, x G. Thus G L = {g L g G} is the left regular representation of G. Throughout the thesis a permutation group X Sym(G) with G L X will be referred to as an overgroup of G L in Sym(G). We shall use the following properties of blocks (see [13, Lemma 3.1]). Proposition Let A be a transitive permutation group on a set Ω which contains a regular abelian subgroup G, and let B be a block system of A. Then for every block B B, G A B is faithful and regular on B, and GA B /A B = G/(G AB ) is regular on B. The following fact follows easily from the above proposition.

20 Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, Corollary Let A be an overgroup of G L in Sym(G), where G is an abelian group. Then every block of A must be an H-coset for some subgroup H G. We shall make use of the following simple observation. Proposition Let A be an overgroup of G L in Sym(G) and T be an orbit of the point stabilizer A e of the identity e of G in A. Then the subgroup {g G gt = T } G is a block of A. Proof. Set H = {g G gt = T }. Consider the set-wise stabilizer A {T } of T in A. For a A we can write a = g L a 1, where g L G L and a 1 A e. Thus a is in A {T } exactly when T = a(t ) = (g L a 1 )T = g L (a 1 (T )) = g L (T ) = gt. Therefore A {T } = H L A e. Since A e H L A e, the orbit e H L A e is a block of A (see [4, Theorem 1.5A]). As e H L A e = e Ae H L = H, the proposition follows. For a subset S G and integer i we let S (i) = {s i s S}. The following result is a special case of [18, part (a) of Theorem 23.9]. Theorem Let A be an overgroup of G L in Sym(G), where G is an abelian group, and let T be an orbit of the point stabilizer A e of the identity e of G in A. Then for every integer i with (i, G ) = 1, T (i) is also an orbit of A e Wreath products of groups Suppose we have a connected graph X = (V, E) with automorphism group G. Consider the disconnected graph i X i consisting of n components X i = (V i, E i ) each isomorphic to X. Then each automorphism of i X i permutes the components. Those automorphisms which fix the components are determined by n automorphisms, automorphism g i G in the i th component. In general such an automorphism will be followed by a permutation π of the components. So letting (v, i) denote a vertex in the i th component, the automorphism with data (π; g 1,..., g n ) will act via (π; g 1,..., g n )(v, i) = (g i (v), π(i))

21 Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, and the composition rule will be given by (π ; g 1,..., g n)(π; g 1,..., g n ) = (π π; g π(1)g 1,..., g π(n)g n ) This calculation motivates the more general wreath product of groups, in which, instead of allowing the full permutation group Sym(n) of the indices of the components, we restrict to a subgroup, or, equivalently, a group with a faithful action on {1,..., n}. Formally, the wreath product of a permutation group Π Sym(n) and a group G is denoted by Π G and consists of the Π G n elements of the form (π; g 1,..., g n ) and multiplication defined by the composition rule above. An important special case is that the permutation group π arises from a group acting on itself by the left regular action. It is not hard to modify the disjoint union of graphs to obtain a graph whose automorphism group realizes the wreath product. The most direct construction is the wreath product of graphs. Given two graphs (digraphs) X 1 and X 2, the wreath product X 1 X 2 is the graph (digraph) with vertex set V (X 1 ) V (X 2 ), and an edge (arc) from (x 1, x 2 ) to (y 1, y 2 ) if and only if (x 1, y 1 ) is an arc of X 1, or x 1 = y 1 and (x 2, y 2 ) is an arc of X 2. For example, in Figure 2.7 we find 8 copies of the tetrahedron graph arranged Figure 2.7: The wreath product of the graph of the tetrahedron, T, with the graph of the cube, C, giving C T. in a wreath product. As observed above, the individual tetrahedron graphs can be independently acted on by their automorphism groups, however, they can now only

22 Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, be permuted by a permutation which respects the incidences of the other factor, the cube graph. Clearly, the automorphism group of C T is Aut(C) Aut(T ). In general, the wreath product Aut(X 1 ) Aut(X 2 ) of the groups Aut(X 1 ) and Aut(X 2 ) acts as a group of automorphisms of the wreath product X 1 X 2 of the graphs X 1 and X 2. It may happen, say in the cases of complete graphs, that Aut(X 1 ) Aut(X 2 ) is only a proper subgroup of the full automorphism group. Here we follow the convention that Aut(X 1 ) Aut(X 2 ) has active group Aut(X 1 ) and passive group Aut(X 2 ), so the order of Aut(X 1 ) Aut(X 2 ) satisfies Aut(X 1 ) Aut(X 2 ) = Aut(X 1 ) Aut(X 2 ) X1. The asymmetry of the wreath product, that is the fact that it is not necessarily true that A B = B A, is illustrated in Figure 2.8 which depicts T C. Figure 2.8: The wreath product T C. A less cluttered construction we call the reduced wreath product, X 1 r X 2. It has vertex set V (X 1 ) (V (X 1 ) V (X 2 )) and arcs (x 1, x 1) if (x 1, x 1) is an arc of X 1, ((x 1, x 2 ), (x 1, y 2 )) if (x 2, y 2 ) is an arc of X 2, and ((x 1, x 2 ), x 1 ) for all x 1 V (X 1 ) and x 2 V (X 2 ). At the cost of V (X 1 ) more vertices the reduced wreath product realizes the wreath product of the automorphism groups with only E(X 1 ) + V (X 1 ) E(X 2 ) + V (X 1 ) V (X 2 )

23 Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, edges instead of V (X 1 ) E(X 2 ) + E(X 1 ) V (X 2 ) 2 for the ordinary wreath product of graphs. The reduced wreath products of the examples C and T as well as T and C are illustrated in Figures 2.9 and 2.10 respectively. Figure 2.9: The reduced wreath product C r T. Figure 2.10: The reduced wreath product T r C. 2.3 Cayley graphs Given a group G and a generating set S, the Cayley graph on G with respect to S is the group with vertex set G and arc set {(g, gs) g G, s S}. The Cayley graph is a directed graph, if S S 1. If the set S is a generates the group, the Cayley graph is connected, otherwise it is disconnected, and its components are isomorphic to Cay( S, S). Sometimes the generating set S is called the connection set. The absolute simplest Cayley graphs to describe and to analyze are those corresponding to cyclic groups, see Figure 2.11 for some Cayley graphs of Z 12.

24 Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, Figure 2.11: Cayley graphs for Z 12 with connection sets consisting of {1}, {1, 11}, and {3, 4}. An important feature of Cayley graphs is that the left regular action on group preserves the arcs, so the group not only acts as a group of automorphisms of the Cayley graph, but acts freely and transitively on the set of vertices. The converse of this observation is also true, that if a group acts as a group of automorphisms on a connected graph such that the action is free and transitive on the vertex set, that graph is a Cayley graph for that group. Free and transitive actions are a fruitful source of Cayley graphs, for example the free and transitive action on the flags of the cube of the example in Figure 2.6 can be used to create a Cayley graph of the symmetry group of the cube, with one vertex for each flag, and the connection set consisting of reflections across the boundary lines of a fixed triangle. Figure 2.12: A Cayley graph constructed from a free transitive action of the symmetry group of the cube on the set of triangular regions subdividing each of the six faces.

25 Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, Arc transitive Cayley digraphs on Cyclic groups Let X be a Cayley digraph Cay(G, S) on a group G. In case S = S 1 we shall regard X also as an undirected graph. It is immediate to see that the left regular representation G L is a group of automorphisms of X acting regularly on the vertices. Denote by Aut(X) e the stabilizer of the identity e of G in Aut(X), and let x Aut(X)e denote the orbit of x under Aut(X). The orbital digraphs of Aut(X) are the Cayley digraphs Cay(G, x Aut(X)e ), x G. Clearly, all orbital digraphs of Aut(X) are arctransitive. Now suppose that G is an abelian group, let H be a subgroup H G. The quotient digraph X/H is the simple digraph where the vertex set is the factor group G/H, and (Hx, Hy) is an arc if and only if X has an arc (h 1 x, h 2 y) for some h 1, h 2 H. Note that, we do not allow X/H to have loops and multiple edges. Note also that X/H is a Cayley digraph on the group G/H, and it is given as X/H = Cay(G/H, S/H), where S/H = {Hs s S}. The following classification of arc-transitive Cayley digraphs on Z 2 n can be deduced from [11, Theorem 4] (for a classification of all arc-transitive Cayley digraphs on cyclic groups, see also the papers [14, 17]). Theorem Let X = Cay(Z 2 n, S) be a connected Cayley digraph. Then X is arc-transitive if and only if S is one of the following sets: Z 2 n \ H, Hz, Hz Hz 1, or Hz Hz 2n 1 1 with n 2, where z is a generator of Z 2 n and H is a subgroup of Z 2 n, 1 H < Z 2 n CI-Graphs The problem of when a Cayley graph can have a different representation over the same group, that is Cay(G, S) = Cay(G, T ), is called the Cayley Representation problem. It is easy to see that the cycle C n can be realized as a Cayley graph over the group Z n. However, there are a number of different Cayley representations over Z n

26 Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, of this graph. That is, the different choices for the generating set are determined by n and the prime factors of n. In fact, Cay(Z n, {k, k 1}) = C n whenever k = Z n. This motivates the notion of equivalent Cayley representations. In fact, two Cayley representations are equivalent if an automorphism of the vertex set maps one generating set onto the other. By coloring the edges consistently with respect to their generators, this is easy to visualize. Equivalent representations of Cayley graphs are those which can be colored the same way. For example, the graph Cay(Z 8, {1, 7, 4}) has two generators of order 8 and one of order 2, and Z 8 has only one other candidate for a generator of order 8, but no other choice for a generator of order 2, so there are only two Cayley representations of this graph, and they are equivalent. One might be tempted to assume that any two representations of a circulant are equivalent. This was the subject of a conjecture of Ádám [1] in the sixties disproved by Elspas and Turner [6] some years later. Take, for instance, the circulant Cay(Z 16, {1, 2, 7, 9, 14, 15}) and Cay(Z 16, {2, 3, 5, 11, 13, 14}). The generating sets are not equivalent, and as seen in the figure below, the isomorphism maps green onto both red an blue, so the representations are not equivalent. This (disproved) conjecture is what spurred the research of so called Cayley Isomorphism graphs or CI-graphs, that is, Cayley graphs whose representations are all equivalent Figure 2.13: This isomorphism shows that Cay(Z 16, {1, 2, 7, 9, 14, 15}) and Cay(Z 16, {2, 3, 5, 11, 13, 14}) are inequivalent representations of a circulant. CI-graphs are of interest primarily because they are useful in discovering isomorphism classes of Cayley graphs. To determine whether two graphs are isomorphic is

27 Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, of course harder than to determine whether two generating sets are equivalent, so such classes are useful in the study of Cayley graphs.

28 Chapter 3 Non-isomorphic groups with isomorphic Cayley graphs. Given a finitely generated group G there will be a Cayley graph associated with every finite generating set S of G. In general, a given group will have many non-isomorphic Cayley graphs, even for generating sets with same cardinality. By contrast, given a graph, we can ask what conditions ensure, in the first place, that the graph is the Cayley graph of a group, and, in the second place, that that realization is unique. It is this second question which concerns us in this thesis, and in this chapter we will give a brief overview of what is known. The answer to the existence part of the question has been answered classically by Sabidussi: Theorem (Sabidussi). A graph X is a Cayley graph if and only if a subgroup G of its automorphism group acts freely and transitively on the vertex set. If the free transitive subgroup is the whole automorphism group of the graph, then that graph is uniquely describable as a Cayley graph. On the other hand, if a free transitive subgroup G Aut(X), so necessarily the vertex stabilizers in Aut(X) are non-trivial, it may or may not be possible to find a non-isomorphic subgroup of Aut(X) which also acts freely and transitively on V (X). In the next section we will give some examples of graphs which can be realized as Cayley Graphs on at least two (non-isomorphic) groups. 22

29 Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, Infinite graphs and free groups An interesting case is that of a free group. If F is a free group freely generated by a set S, the Cayley graph of F with respect to S is an infinite 2 S valent tree. The vertex stabilizers in the automorphism group of a 2 S valent tree are infinite, so it might be supposed that there are many opportunities for the existence of a different free transitive subgroup, however a theorem of Serre states: Theorem Every group which acts freely on a tree is free. Note that to act freely on the tree the group must act freely on both the vertices and the edges. If you relax that condition on the edges, so one allows edges to be fixed, then, since the action is still free on the vertices, the endpoints of a fixed edge must be interchanged under an automorphism. Such an action on a tree may be free, and in fact transitive on the vertices, yielding a realization of the tree as a Cayley graph of a non-free group, see Figure 3.1. Note that the upper graph is b a c Figure 3.1: The lower digraph is part of an infinite 2-valent tree, freely acted on by the free rank 1 infinite cycle group of translations generated by c. The upper graph is acted on by a group isomorphic to the free product Z 2 Z 2 generated by the involutory reflections a and b. drawn as a simple graph and the lower graph is drawn as a digraph, so to establish the isomorphism we would either disregard the directions in the digraph or regard the undirected edges in the upper graph as matched pairs of directed edges, and add c 1 to the generating set for the lower Cayley graph. In general we will not point out such obvious modifications. Another treelike example is illustrated in Figure 3.2 where the Cayley graphs are isomorphic, with the isomorphism preserving the directions on the edges, and yet the groups can be shown to be non-isomorphic. The group on the left has elements of order three, and is in fact clearly the free product Z 3 Z 3 Z 3. The group on the

30 Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, Figure 3.2: The simple digraph on the left depicts a small section of the infinite Cayley graph associated to the group with presentation a, b, c a 3 = b 3 = c 3 = 1. The recolored digraph the right depicts the Cayley graph of the group with presentation a, b, c abc = 1. right has all non-trivial elements of infinite order. In fact, the group on the right is a free group, freely generated by, say, a and b, with the equation abc = 1 merely identifying the third generator c redundantly as the element c = b 1 a 1. In terms of the graph, if on deletes the red edges corresponding to the redundant generator c, the result is an infinite four valent tree. The previous examples suggest that examples may be found by considering graphs which are not only vertex transitive, but edge transitive as well, for instance, the graphs of plane lattices and platonic solids. The graph of the integer square lattice with edges oriented counterclockwise around every square face has two distinct actions which are free and transitive on the vertices, see Figure 3.3. The group of the action on the left is generated by two 90 rotations a and b. The group of the Cayley graph on the right is generated by two perpendicular glide reflections of the lattice. So as transformation groups they are not the same, however are they non-isomorphic? The technique of the previous example is not sufficient since these two groups have the same abelianizations. In the abelianization of a, b (ab 1 ) 2 = (ab) 2 = 1 both a and b are of order 4, since 1 = (ab 1 ) 2 (ab) 2 = a 4 and 1 = (ab 1 ) 2 (ab) 2 = b 4, while in the abelianization a, b a 4 = b 4 = (ab 1 ) 2 = 1 we have (ab) 2 = (ab) 2 b 4 = (ab 1 ) 2 = 1. So both have abelianization Z 4 Z 2 = a ab 1.

31 Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, z y a b a b 1 x Figure 3.3: Two Cayley graphs on the graph of the integer plane lattice. The group on the left is presented by a, b a 4 = b 4 = (ab 1 ) 2 = 1, and the group on the right is presented by a, b (ab 1 ) 2 = (ab) 2 = 1. Nevertheless, we can use the action of the group on the lattice to show that a, b (ab 1 ) 2 = (ab) 2 = 1 has no element of order four, even though its abelianization does have such elements. Every element in a, b (ab 1 ) 2 = (ab) 2 = 1 corresponds to a path from 1 along the colored integer lattice ending in a vertex whose directed edges in the lattice are in one of four possible configurations, indicated by 1, x, y or z in Figure 3.3, depending on whether the green arrows are oriented to the left or the right, and whether the red arrows are oriented up or down. If the path from 1 ends at another vertex (n, m) of the same type as 1, then repeating that path will result in another vertex (2n, 2m), also of type 1, and so on, with that element of infinite order. If the path from 1 ends at a vertex of type y, so the orientations are exactly reversed, then repeating that vertex will return to 1, and the element is of order 2. If the path from 1 ends at a vertex (n, m) of type x, with the same red orientation and reversed green orientation, then repeating that path starting at x will result in (2n, 0), continuing to (3n, m), etc, so again we have an element of infinite order. In the same way, a the path from 1 which ends at a vertex of type y corresponds to a group element of infinite order. So all elements of a, b (ab 1 ) 2 = (ab) 2 = 1 have order 1 or 2 or. Geometrically, the elements act on the lattice by glide reflections (types x and z), half-turns (type z) and translations (type 1). The triangular lattice with all triangles oriented counterclockwise can also be

32 Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, colored in two ways to form Cayley graphs, see Figure 3.4. Here it is fairly easy Figure 3.4: Two Cayley graphs on the graph of the triangular plane lattice. On the left the group has presentation a, b, c a 3 = b 3 = c 3 = abc = 1 while on the right the group has presentation a, b, c abc = bca = 1 to show that the underlying groups are non-isomorphic. In the group on the left the generator a is redundant, the equation abc = 1 identifies a with (bc) 1 and the remaining equation cba = 1 states 1 = cb(bc) 1 = cbc 1 b 1, in other words, the only relation between b and c is that they commute, so a, b, c abc = bca = 1 = b, c bc = cb = 1 = b c = Z Z a free abelian group of rank 2, having no elements of order 3. The graph of the regular hexagonal lattice has two colorings as Cayley graphs in which all the undirected edges refer to involutions, see Figure 3.5. The groups may be shown to be non-isomorphic, again by considering their abelianizations. In the abelianization of a, b, c a 2 = b 2 = c 2 = (ab) 3 = (bc) 3 = (ca) 3 = 1 the relation 1 = (ab) 3 = ababab = a 3 b 3 = ab says simply that generators a and b are equal. The three relations (ab) 3 = (bc) 3 = (ca) 3 = 1 therefore imply that a = b = c, and the group is isomorphic to Z 2. For the abelianization of a, b, c a 2 = b 2 = c 2 = (abc) 2 = 1 the relation (abc) 2 = 1 is a consequence of the other three, so the group is isomorphic to Z 2 Z 2 Z 2. There is another coloring of graph of the hexagonal lattice in which only one of the generators is an involution, see Figure 3.6. Writing c = ab, this is equivalent to a, c a 2 = (ac) 6 = c 3 = 1, with the relation (ac) 6 = 1 a consequence of the other

33 Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, b a b a c c Figure 3.5: Two Cayley graphs on the graph of the regular hexagonal lattice. The group on the left is presented by a, b, c a 2 = b 2 = c 2 = (ab) 3 = (bc) 3 = (ca) 3 = 1 and the group on the right is presented by a, b, c a 2 = b 2 = c 2 = (abc) 2 = 1. b a Figure 3.6: Another Cayley graphs on the graph of the regular hexagonal lattice. The group is presented by a, b a 2 = b 6 = (ab) 3 = 1. two in the abelianization, so the group abelianizes to Z 2 Z 3, and the graph of the hexagonal lattice is the Cayley graph of three non-isomorphic groups. 3.2 Graphs of the Platonic solids There are five Platonic solid graphs, corresponding to the tetrahedron, the cube, the octahedron, the icosahedron, and the dodecahedron. Four of them realize Cayley graphs of non-isomorphic groups. For the graph of the tetrahedron, see Figure 3.7, the vertices each have valence 3, so for the graph to be a Cayley graph, the generators for the group would have to either be three involutions, partitioning the edges into three perfect matchings, which decomposition corresponds to the Cayley graph

34 Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, Figure 3.7: Two realizations of the tetrahedral graph as a Cayley graph. On the left the Klein four group acts freely and transitively on the vertex set by three 180 rotations along the three coordinate axis. On the right a 90 rotary reflection about an axis through the centroids of one pair of antipodal edges generates a cyclic group isomorphic to Z 4 acting freely and transitively on the vertices. The second generator is the square of the first. on the left of the figure; or the generators consist of one involution and one noninvolution. Deleting a perfect matching of two edges from the tetrahedron graph yields a simple 4-cycle, so the only other possibility is an involution and an element of order four, as see on the right of the Figure. We may similarly analyze the possible realizations of the graph of the cube as a Cayley graph. Like the tetrahedron, each vertex has valence 3, so there could be either one or three involutions. If the involution corresponds to a parallel class of edges, say the red edges of Figure 3.8, The eight remaining edges are partitioned into two 4-cycles. If there are two oriented 4-cycles, necessarily belonging to the same generator, either the cycles have the same or contrasting orientations, illustrated in the left two Cayley graphs of Figure 3.8, in which case the group will be either isomorphic to the direct product Z 4 Z 2 or the dihedral group D 8. The edges of the two 4-cycles may also correspond to involutions, such as the Cayley graph on the right of Figure 3.8, giving a group isomorphic to Z 3 2. There is yet another possibility for the cube. There is a second type of perfect matching in the graph of the cube, illustrated by the red edges of Figure 3.9. In this case the involution corresponding the red edge cannot be a rotation, since that would not preserve the set of red edges, and that transformation must be a